SEQUENCES AND SERIES Sequences and Summation Notation

Transcription

1 Zia Soleil /Icoica/ The Getty Images C H A P T E R SEQUENCES AND SERIES. Sequeces ad Summatio Notatio. Arithmetic Sequeces.3 Geometric Sequeces.4 Mathematics of Fiace.5 Mathematical Iductio.6 The Biomial Theorem FOCUS ON MODELING Modelig with Recursive Sequeces A sequece is a list of umbers writte i a specific order. For example, the height that a boucig ball reaches after each bouce is a sequece. This sequece has a defiite patter; describig the patter allows us to predict the height that the ball reaches after ay umber of bouces. / /4 /8 /6 The amout i a bak accout at the ed of each moth, mortgage paymets, ad the amout of a auity are also sequeces. The formulas that geerate these sequeces drive our ecoomy they allow us to borrow moey to buy our dream home closer to graduatio tha to retiremet. I this chapter we study these ad other applicatios of sequeces. 783 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

2 784 CHAPTER Sequeces ad Series. SEQUENCES AND SUMMATION NOTATION Sequeces Recursively Defied Sequeces The Partial Sums of a Sequece Sigma Notatio Roughly speakig, a sequece is a ifiite list of umbers. The umbers i the sequece are ofte writte as a, a, a 3,... The dots mea that the list cotiues forever. A simple example is the sequece 5, 0, 5, 0, 5,... a 5... We ca describe the patter of the sequece displayed above by the followig formula: You may have already thought of a differet way to describe the patter amely, you go from oe umber to the ext by addig 5. This atural way of describig the sequece is expressed by the recursive formula: startig with a 5. Try substitutig,, 3,... i each of these formulas to see how they produce the umbers i the sequece. I this sectio we see how these differet ways are used to describe specific sequeces. Sequeces a a a 3 a 5 a a 5 Ay ordered list of umbers ca be viewed as a fuctio whose iput values are,, 3,... ad whose output values are the umbers i the list. So we defie a sequece as follows: a 4 DEFINITION OF A SEQUENCE A sequece is a fuctio f whose domai is the set of atural umbers. The terms of the sequece are the fuctio values f, f, f 3,..., f,... We usually write a istead of the fuctio otatio f. So the terms of the sequece are writte as a, a, a 3,..., a,... The umber a is called the first term, a is called the secod term, ad i geeral, a is called the th term. Aother way to write this sequece is to use fuctio otatio: a so a, a 4, a3 6,... Here is a simple example of a sequece:, 4, 6, 8, 0,... We ca write a sequece i this way whe it s clear what the subsequet terms of the sequece are. This sequece cosists of eve umbers. To be more accurate, however, we eed to specify a procedure for fidig all the terms of the sequece. This ca be doe by givig a formula for the th term a of the sequece. I this case, a Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

3 SECTION. Sequeces ad Summatio Notatio 785 ad the sequece ca be writte as, 4, 6, 8,...,,... st term d term 3rd term 4th term th term Notice how the formula a gives all the terms of the sequece. For istace, substitutig,, 3, ad 4 for gives the first four terms: To fid the 03rd term of this sequece, we use 03 to get EXAMPLE Fidig the Terms of a Sequece Fid the first five terms ad the 00th term of the sequece defied by each formula. (a) (c) a t a # a # 4 a 3 # 3 6 a4 # 4 8 a 03 # (b) c (d) r SOLUTION To fid the first five terms, we substitute,, 3, 4, ad 5 i the formula for the th term. To fid the 00th term, we substitute 00. This gives the followig. th term First five terms 00th term (a),3,5,7,9 99 (b) 0, 3, 8, 5, (c) (d), 3, 3 4, 4 5, 5 6, 4, 8, 6, a Terms are decreasig NOW TRY EXERCISES 3, 5, 7, AND9 0 FIGURE I Example (d) the presece of i the sequece has the effect of makig successive terms alterately egative ad positive. It is ofte useful to picture a sequece by sketchig its graph. Sice a sequece is a fuctio whose domai is the atural umbers, we ca draw its graph i the Cartesia plae. For istace, the graph of the sequece a Terms alterate i sig,, 3, 4, 5, 6,...,,... 0 _ FIGURE 3 5 is show i Figure. Compare the graph of the sequece show i Figure to the graph of,, 3, 4, 5, 6,...,,... show i Figure. The graph of every sequece cosists of isolated poits that are ot coected. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

4 786 CHAPTER Sequeces ad Series Graphig calculators are useful i aalyzig sequeces. To work with sequeces o a TI-83, we put the calculator i Seq mode ( sequece mode) as i Figure 3(a). If we eter the sequece u / of Example (c), we ca display the terms usig the TABLE commad as show i Figure 3(b). We ca also graph the sequece as show i Figure 3(c)..5 FIGURE 3 u / Plot Plot Plot3 Mi= u( ) = /( +) (a) u( ) = (b) FIGURE (c) Not all sequeces ca be defied by a formula. For example, there is o kow formula for the sequece of prime umbers:*, 3, 5, 7,, 3, 7, 9, 3,... Large Prime Numbers The search for large primes fasciates may people. As of this writig, the largest kow prime umber is 43,,609 It was discovered by Edso Smith of the Departmet of Mathematics at UCLA. I decimal otatio this umber cotais,978,89 digits. If it were writte i full, it would occupy more tha three times as may pages as this book cotais. Smith was workig with a large Iteret group kow as GIMPS (the Great Iteret Mersee Prime Search). Numbers of the form p, where p is prime, are called Mersee umbers ad are more easily checked for primality tha others. That is why the largest kow primes are of this form. Fidig patters is a importat part of mathematics. Cosider a sequece that begis Ca you detect a patter i these umbers? I other words, ca you defie a sequece whose first four terms are these umbers? The aswer to this questio seems easy; these umbers are the squares of the umbers,, 3, 4. Thus, the sequece we are lookig for is defied by a. However, this is ot the oly sequece whose first four terms are, 4, 9, 6. I other words, the aswer to our problem is ot uique (see Exercise 80). I the ext example we are iterested i fidig a obvious sequece whose first few terms agree with the give oes. EXAMPLE Fidig the th Term of a Sequece Fid the th term of a sequece whose first several terms are give. (a), 3 4, 5 6, 7 8,... (b), 4, 8, 6, 3,... SOLUTION (a) We otice that the umerators of these fractios are the odd umbers ad the deomiators are the eve umbers. Eve umbers are of the form, ad odd umbers are of the form (a odd umber differs from a eve umber by ). So a sequece that has these umbers for its first four terms is give by a (b) These umbers are powers of, ad they alterate i sig, so a sequece that agrees with these terms is give by You should check that these formulas do ideed geerate the give terms. NOW TRY EXERCISES 5, 7, AND9, 4, 9, 6,... a Recursively Defied Sequeces Some sequeces do ot have simple defiig formulas like those of the precedig example. The th term of a sequece may deped o some or all of the terms precedig it. A sequece defied i this way is called recursive. Here are two examples. * A prime umber is a whole umber p whose oly divisors are p ad. (By covetio, the umber is ot cosidered prime.) Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

5 SECTION. Sequeces ad Summatio Notatio 787 ERATOSTHENES (circa B.C.) was a reowed Greek geographer, mathematicia, ad astroomer. He accurately calculated the circumferece of the earth by a igeious method. He is most famous, however, for his method for fidig primes, ow called the sieve of Eratosthees. The method cosists of listig the itegers, begiig with (the first prime), ad the crossig out all the multiples of, which are ot prime. The ext umber remaiig o the list is 3 (the secod prime), so we agai cross out all multiples of it. The ext remaiig umber is 5 (the third prime umber), ad we cross out all multiples of it, ad so o. I this way all umbers that are ot prime are crossed out, ad the remaiig umbers are the primes EXAMPLE 3 Fidig the Terms of a Recursively Defied Sequece Fid the first five terms of the sequece defied recursively by a ad SOLUTION The defiig formula for this sequece is recursive. It allows us to fid the th term a if we kow the precedig term a. Thus, we ca fid the secod term from the first term, the third term from the secod term, the fourth term from the third term, ad so o. Sice we are give the first term a, we ca proceed as follows. Thus the first five terms of this sequece are NOW TRY EXERCISE 3 a 3a a 3a 3 9 a 3 3a a 4 3a a 5 3a , 9, 33, 05, 3,... Note that to fid the 0th term of the sequece i Example 3, we must first fid all 9 precedig terms. This is most easily doe by usig a graphig calculator. Figure 4(a) shows how to eter this sequece o the TI-83 calculator. From Figure 4(b) we see that the 0th term of the sequece is a 0 4,649,045,865. Plot Plot Plot3 Mi= u( )=3(u( -)+) u( Mi)={} u(0) FIGURE 4 (a) u 3u, u (b) EXAMPLE 4 The Fiboacci Sequece Fid the first terms of the sequece defied recursively by F, F ad F F F The Grager Collectio, New York FIBONACCI (75 50) was bor i Pisa, Italy, ad was educated i North Africa. He traveled widely i the Mediterraea area ad leared the various methods the i use for writig umbers. O returig to Pisa i 0, Fiboacci advocated the use of the Hidu-Arabic decimal system, the oe we use today, over the Roma umeral system that was used i Europe i his time. His most famous book, Liber Abaci, expouds o the advatages of the Hidu-Arabic umerals. I fact, multiplicatio ad divisio were so complicated usig Roma umerals that a college degree was ecessary to master these skills. Iterestigly, i 99 the city of Florece outlawed the use of the decimal system for merchats ad busiesses, requirig umbers to be writte i Roma umerals or words. Oe ca oly speculate about the reasos for this law. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

6 788 CHAPTER Sequeces ad Series SOLUTION To fid F, we eed to fid the two precedig terms, F ad F. Sice we are give F ad F, we proceed as follows. F 3 F F F 4 F 3 F 3 F 5 F 4 F It s clear what is happeig here. Each term is simply the sum of the two terms that precede it, so we ca easily write dow as may terms as we please. Here are the first terms:,,, 3, 5, 8, 3,, 34, 55, 89,... NOW TRY EXERCISE 7 The sequece i Example 4 is called the Fiboacci sequece, amed after the 3th cetury Italia mathematicia who used it to solve a problem about the breedig of rabbits (see Exercise 79). The sequece also occurs i umerous other applicatios i ature. (See Figures 5 ad 6.) I fact, so may pheomea behave like the Fiboacci sequece that oe mathematical joural, the Fiboacci Quarterly, is devoted etirely to its properties FIGURE 5 The Fiboacci sequece i the brachig of a tree FIGURE 6 Fiboacci spiral Nautilus shell Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

7 The Partial Sums of a Sequece SECTION. Sequeces ad Summatio Notatio 789 I calculus we are ofte iterested i addig the terms of a sequece. This leads to the followig defiitio. THE PARTIAL SUMS OF A SEQUENCE For the sequece the partial sums are a, a, a 3, a 4,..., a,... S a S a a S 3 a a a 3 S 4 a a a 3 a 4. S a a a 3... a. S is called the first partial sum, S is the secod partial sum, ad so o. S is called the th partial sum. The sequece S, S, S 3,...,S,... is called the sequece of partial sums. EXAMPLE 5 Fidig the Partial Sums of a Sequece Fid the first four partial sums ad the th partial sum of the sequece give by a /. SOLUTION The terms of the sequece are The first four partial sums are, 4, 8,... S Partial sums of the sequece S S S Sfi S 4 S S a a a Terms of the sequece a afi FIGURE 7 Graph of the sequece a ad the sequece of partial sums S Notice that i the value of each partial sum, the deomiator is a power of ad the umerator is oe less tha the deomiator. I geeral, the th partial sum is The first five terms of a ad S are graphed i Figure 7. NOW TRY EXERCISE 37 S S Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

8 790 CHAPTER Sequeces ad Series EXAMPLE 6 Fidig the Partial Sums of a Sequece Fid the first four partial sums ad the th partial sum of the sequece give by SOLUTION a The first four partial sums are S a b S a b a 3 b 3 S 3 a b a 3 b a 3 4 b 4 S 4 a b a 3 b a 3 4 b a 4 5 b 5 Do you detect a patter here? Of course. The th partial sum is NOW TRY EXERCISE 39 S This tells us to add This tells us to ed with k a a k k Sigma Notatio Give a sequece a, a, a 3, a 4,... we ca write the sum of the first terms usig summatio otatio, or sigma otatio. This otatio derives its ame from the Greek letter (capital sigma, correspodig to our S for sum ). Sigma otatio is used as follows: a a k a a a 3 a 4... a k This tells us to start with k The left side of this expressio is read, The sum of a k from k to k. The letter k is called the idex of summatio, or the summatio variable, ad the idea is to replace k i the expressio after the sigma by the itegers,, 3,...,, ad add the resultig expressios, arrivig at the right side of the equatio. EXAMPLE 7 Sigma Notatio Fid each sum (a) a k (b) a (c) a i (d) k j3 j i5 SOLUTION (a) 5 a k k 5 (b) a j3 j a i Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

9 SECTION. Sequeces ad Summatio Notatio 79 sum(seq(k,k,,5,)) 55 sum(seq(/j,j,3,5, )) Frac 47/60 (c) (d) 0 a i i5 6 a i FIGURE 8 We ca use a graphig calculator to evaluate sums. For istace, Figure 8 shows how the TI-83 ca be used to evaluate the sums i parts (a) ad (b) of Example 7. NOW TRY EXERCISES 4 AND 43 EXAMPLE 8 Writig Sums i Sigma Notatio Write each sum usig sigma otatio. (a) (b) SOLUTION (a) We ca write (b) A atural way to write this sum is a a 77 However, there is o uique way of writig a sum i sigma otatio. We could also write this sum as a k 3 k0 k 3 k k3 k or a k NOW TRY EXERCISES 6 AND 63 k The Golde Ratio The aciet Greeks cosidered a lie segmet to be divided ito the golde ratio if the ratio of the shorter part to the loger part is the same as the ratio of the loger part to the whole segmet. x Thus the segmet show is divided ito the golde ratio if The golde ratio is related to the Fiboacci sequece. I fact, it ca be show by usig calculus* that the ratio of two successive Fiboacci umbers F F gets closer to the golde ratio the larger the value of.try fidig this ratio for 0. x x x This leads to a quadratic equatio whose positive solutio is x 5.68 This ratio occurs aturally i may places. For istace, psychological experimets show that the most pleasig shape of rectagle is oe whose sides are i golde ratio.the aciet Greeks agreed with this ad built their temples i this ratio. Clark Dubar/Corbis.68 *See Priciples of Problem Solvig 3 at the book compaio website: Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

10 79 CHAPTER Sequeces ad Series The followig properties of sums are atural cosequeces of properties of the real umbers. PROPERTIES OF SUMS Let a, a, a 3, a 4,... ad b, b, b 3, b 4,... be sequeces. The for every positive iteger ad ay real umber c, the followig properties hold... a a k b k a a k a k a a k b k a a k a k k k 3. a ca k c a a a k b k k b k k b k k PROOF To prove Property, we write out the left side of the equatio to get a a k b k a b a b a 3 b 3... a b Because additio is commutative ad associative, we ca rearrage the terms o the right side to read k a a k b k a a a 3... a b b b 3... b k Rewritig the right side usig sigma otatio gives Property. Property is proved i a similar maer. To prove Property 3, we use the Distributive Property: a ca k ca ca ca 3... ca k ca a a 3... a c a a k a k b. EXERCISES CONCEPTS. A sequece is a fuctio whose domai is.. The th partial sum of a sequece is the sum of the first terms of the sequece. So for the sequece a the fourth partial sum is S 4. SKILLS 3 Fid the first four terms ad the 00th term of the sequece. 3. a 4. a 3 5. a 6. a 7. a 8. a 9. a 0. a. a. a Fid the first five terms of the give recursively defied sequece. 3. a a ad a 3 4. a a ad a 8 5. a a ad a 6. a ad a a 7. a a a ad a, a 8. a a a a 3 ad a a a 3 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

11 SECTION. Sequeces ad Summatio Notatio Use a graphig calculator to do the followig. (a) Fid the first 0 terms of the sequece. (b) Graph the first 0 terms of the sequece. 9. a a. a. a 4 3. a a ad a 4. a a a ad a, a Fid the th term of a sequece whose first several terms are give. 5.,4,8,6, , 9, 8,... 7.,4,7,0, , 5, 5, 65, , 3 4, 5 9, 6, 7 5, , 4 5, 5 6, 6 7, ,,0,,0,,... 3.,, 3, 4, 5, 6, Fid the first six partial sums S, S, S 3, S 4, S 5, S 6 of the sequece. 33., 3, 5, 7, ,,3,4, ,,,,... 3, 3, 3 3, 3 4, Fid the first four partial sums ad the th partial sum of the sequece a. 37. a , a 39. a 40. a log a [Hit: Use a property of logarithms to b write the th term as a differece.] Write the sum without usig sigma otatio. 55. a k a k k 59. a x Write the sum usig sigma otatio k 6 k0 00 k3 l 3 l 3 4 l 4 5 l l 00 # # 3 3 # x x x 3...x x 3x 4x 3 5x x Fid a formula for the th term of the sequece,, 3, 43,... [Hit: Write each term as a power of.] 70. Defie the sequece G a kk # a 5 5 b Use the TABLE commad o a graphig calculator to fid the first 0 terms of this sequece. Compare to the Fiboacci sequece F. 4 a i0 9 k6 a j i i j j x 4 48 Fid the sum a k 4. k a 44. k k a 3 i i 5 k k 47. a Use a graphig calculator to evaluate the sum. 0 k k 49. a a j j 5. j a 54. a 0 4 a k k 00 a j j a 0 i4 3 a i i i 00 a 3k 4 k 5 a j5 j APPLICATIONS 7. Compoud Iterest Julio deposits $000 i a savigs accout that pays.4% iterest per year compouded mothly. The amout i the accout after moths is give by the sequece A 000 a 0.04 b (a) Fid the first six terms of the sequece. (b) Fid the amout i the accout after 3 years. 7. Compoud Iterest Hele deposits $00 at the ed of each moth ito a accout that pays 6% iterest per year compouded mothly. The amout of iterest she has accumulated after moths is give by the sequece I 00 a b (a) Fid the first six terms of the sequece. (b) Fid the iterest she has accumulated after 5 years. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

12 794 CHAPTER Sequeces ad Series 73. Populatio of a City A city was icorporated i 004 with a populatio of 35,000. It is expected that the populatio will icrease at a rate of % per year. The populatio years after 004 is give by the sequece P 35,000.0 (a) Fid the first five terms of the sequece. (b) Fid the populatio i Payig off a Debt Margarita borrows $0,000 from her ucle ad agrees to repay it i mothly istallmets of $00. Her ucle charges 0.5% iterest per moth o the balace. (a) Show that her balace A i the th moth is give recursively by A 0 0,000 ad A.005A 00 (b) Fid her balace after six moths. 75. Fish Farmig A fish farmer has 5000 catfish i his pod. The umber of catfish icreases by 8% per moth, ad the farmer harvests 300 catfish per moth. (a) Show that the catfish populatio P after moths is give recursively by P ad P.08P 300 (b) How may fish are i the pod after moths? 76. Price of a House The media price of a house i Orage Couty icreases by about 6% per year. I 00 the media price was $40,000. Let P be the media price years after 00. (a) Fid a formula for the sequece P. (b) Fid the expected media price i Salary Icreases A ewly hired salesma is promised a begiig salary of $30,000 a year with a $000 raise every year. Let S be his salary i his th year of employmet. (a) Fid a recursive defiitio of S. (b) Fid his salary i his fifth year of employmet. 78. Cocetratio of a Solutio A biologist is tryig to fid the optimal salt cocetratio for the growth of a certai species of mollusk. She begis with a brie solutio that has 4 g/l of salt ad icreases the cocetratio by 0% every day. Let C 0 deote the iitial cocetratio ad C the cocetratio after days. (a) Fid a recursive defiitio of C. (b) Fid the salt cocetratio after 8 days. 79. Fiboacci s Rabbits Fiboacci posed the followig problem: Suppose that rabbits live forever ad that every moth each pair produces a ew pair that becomes productive at age moths. If we start with oe ewbor pair, how may pairs of rabbits will we have i the th moth? Show that the aswer is F, where F is the th term of the Fiboacci sequece. DISCOVERY DISCUSSION WRITING 80. Differet Sequeces That Start the Same (a) Show that the first four terms of the sequece a are (b) Show that the first four terms of the sequece a 3 4 are also (c) Fid a sequece whose first six terms are the same as those of a but whose succeedig terms differ from this sequece. (d) Fid two differet sequeces that begi 8. A Recursively Defied Sequece Fid the first 40 terms of the sequece defied by ad a. Do the same if a 5. Make a cojecture about this type of sequece. Try several other values for a, to test your cojecture. 8. A Differet Type of Recursio Fid the first 0 terms of the sequece defied by with a a c 3a, 4, 9, 6,..., 4, 9, 6,..., 4, 8, 6,... if a is a eve umber if a is a odd umber a a a a a a ad a How is this recursive sequece differet from the others i this sectio?. ARITHMETIC SEQUENCES Arithmetic Sequeces Partial Sums of Arithmetic Sequeces I this sectio we study a special type of sequece, called a arithmetic sequece. Arithmetic Sequeces Perhaps the simplest way to geerate a sequece is to start with a umber a ad add to it a fixed costat d, over ad over agai. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

13 SECTION. Arithmetic Sequeces 795 DEFINITION OF AN ARITHMETIC SEQUENCE A arithmetic sequece is a sequece of the form a, a d, a d, a 3d, a 4d,... The umber a is the first term, ad d is the commo differece of the sequece. The th term of a arithmetic sequece is give by a a d The umber d is called the commo differece because ay two cosecutive terms of a arithmetic sequece differ by d FIGURE EXAMPLE Arithmetic Sequeces (a) If a ad d 3, the we have the arithmetic sequece or Ay two cosecutive terms of this sequece differ by d 3. The th term is a 3. (b) Cosider the arithmetic sequece Here the commo differece is d 5. The terms of a arithmetic sequece decrease if the commo differece is egative. The th term is a 9 5. (c) The graph of the arithmetic sequece a is show i Figure. Notice that the poits i the graph lie o the straight lie y x, which has slope d. NOW TRY EXERCISES 5, 9, AND3, 3, 6, 9,..., 5, 8,,... 9, 4,, 6,,... A arithmetic sequece is determied completely by the first term a ad the commo differece d. Thus, if we kow the first two terms of a arithmetic sequece, the we ca fid a formula for the th term, as the ext example shows. EXAMPLE Fidig Terms of a Arithmetic Sequece Fid the first six terms ad the 300th term of the arithmetic sequece 3, 7,... SOLUTION Sice the first term is 3, we have a 3. The commo differece is d Thus the th term of this sequece is a 3 6 From this we fid the first six terms: 3, 7,, 5,, 7,... The 300th term is a NOW TRY EXERCISE 7 The ext example shows that a arithmetic sequece is determied completely by ay two of its terms. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

14 796 CHAPTER Sequeces ad Series MATHEMATICS IN THE MODERN WORLD Fair Divisio of Assets Dividig a asset fairly amog a umber of people is of great iterest to mathematicias. Problems of this ature iclude dividig the atioal budget, disputed lad, or assets i divorce cases. I 994 Brams ad Taylor foud a mathematical way of dividig thigs fairly. Their solutio has bee applied to divisio problems i political sciece, legal proceedigs, ad other areas. To uderstad the problem, cosider the followig example. Suppose persos A ad B wat to divide a property fairly betwee them. To divide it fairly meas that both A ad B must be satisfied with the outcome of the divisio. Solutio: A gets to divide the property ito two pieces, the B gets to choose the piece he or she wats. Sice both A ad B had a part i the divisio process, each should be satisfied.the situatio becomes much more complicated if three or more people are ivolved (ad that s where mathematics comes i). Dividig thigs fairly ivolves much more tha simply cuttig thigs i half; it must take ito accout the relative worth each perso attaches to the thig beig divided. A story from the Bible illustrates this clearly.two wome appear before Kig Solomo, each claimig to be the mother of the same ewbor baby. Kig Solomo s solutio is to divide the baby i half! The real mother, who attaches far more worth to the baby tha ayoe else does, immediately gives up her claim to the baby to save the baby s life. Mathematical solutios to fairdivisio problems have recetly bee applied i a iteratioal treaty, the Covetio o the Law of the Sea. If a coutry wats to develop a portio of the sea floor, it is required to divide the portio ito two parts, oe part to be used by itself ad the other by a cosortium that will preserve it for later use by a less developed coutry. The cosortium gets first pick. EXAMPLE 3 Fidig Terms of a Arithmetic Sequece The th term of a arithmetic sequece is 5, ad the 9th term is 9. Fid the 000th term. SOLUTION From this formula we get To fid the th term of this sequece, we eed to fid a ad d i the formula Sice a 5 ad a 9 9, we get the two equatios: Solvig this system for a ad d, we get a ad d 5. (Verify this.) Thus the th term of this sequece is The 000th term is a NOW TRY EXERCISE 37 Partial Sums of Arithmetic Sequeces Suppose we wat to fid the sum of the umbers,, 3, 4,...,00, that is, Whe the famous mathematicia C. F. Gauss was a schoolboy, his teacher posed this problem to the class ad expected that it would keep the studets busy for a log time. But Gauss aswered the questio almost immediately. His idea was this: Sice we are addig umbers produced accordig to a fixed patter, there must also be a patter (or formula) for fidig the sum. He started by writig the umbers from to 00 ad the below them wrote the same umbers i reverse order. Writig S for the sum ad addig correspodig terms give It follows that S 000 0,00 ad so S Of course, the sequece of atural umbers,, 3,... is a arithmetic sequece (with a ad d ), ad the method for summig the first 00 terms of this sequece ca be used to fid a formula for the th partial sum of ay arithmetic sequece. We wat to fid the sum of the first terms of the arithmetic sequece whose terms are a k a k d; that is, we wat to fid S a k 3a k d4 a a d a d a 3d... 3a d4 Usig Gauss s method, we write a a d a a d a 0d a 9 a 9 d a 8d 5 a 0d e 9 a 8d a 5 00 a k k S S S S a Óa dô... 3a Ó Ôd4 3a Ó Ôd4 S 3a Ó Ôd4 3a Ó Ôd4... Óa dô a S 3a Ó Ôd4 3a Ó Ôd4... 3a Ó Ôd4 3a Ó Ôd4 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

15 SECTION. Arithmetic Sequeces 797 There are idetical terms o the right side of this equatio, so S 3a d4 S 3a d4 Notice that a a d is the th term of this sequece. So we ca write S 3a a d4 a a a b This last formula says that the sum of the first terms of a arithmetic sequece is the average of the first ad th terms multiplied by, the umber of terms i the sum. We ow summarize this result. PARTIAL SUMS OF AN ARITHMETIC SEQUENCE For the arithmetic sequece a a d the th partial sum S a a d a d a 3d... 3a d4 is give by either of the followig formulas... S a a a S 3a d4 b EXAMPLE 4 Fidig a Partial Sum of a Arithmetic Sequece Fid the sum of the first 40 terms of the arithmetic sequece 3, 7,, 5,... SOLUTION For this arithmetic sequece, a 3 ad d 4. Usig Formula for the partial sum of a arithmetic sequece, we get NOW TRY EXERCISE 43 S EXAMPLE 5 Fidig a Partial Sum of a Arithmetic Sequece Fid the sum of the first 50 odd umbers. SOLUTION The odd umbers form a arithmetic sequece with a ad d. The th term is a, so the 50th odd umber is a Substitutig i Formula for the partial sum of a arithmetic sequece, we get S a a a b 50 a b 50 # NOW TRY EXERCISE 49 EXAMPLE 6 Fidig the Seatig Capacity of a Amphitheater A amphitheater has 50 rows of seats with 30 seats i the first row, 3 i the secod, 34 i the third, ad so o. Fid the total umber of seats. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

16 798 CHAPTER Sequeces ad Series SOLUTION The umbers of seats i the rows form a arithmetic sequece with a 30 ad d. Sice there are 50 rows, the total umber of seats is the sum S Thus the amphitheater has 3950 seats. S 3a d4 NOW TRY EXERCISE 65 Stage EXAMPLE 7 Fidig the Number of Terms i a Partial Sum How may terms of the arithmetic sequece 5, 7, 9,... must be added to get 57? SOLUTION We are asked to fid whe S 57. Substitutig a 5, d, ad S 57 i Formula for the partial sum of a arithmetic sequece, we get 57 3 # 5 4 S 3a d Distributive Property Expad 0 6 Factor This gives or 6. But sice is the umber of terms i this partial sum, we must have. NOW TRY EXERCISE 59. EXERCISES CONCEPTS. A arithmetic sequece is a sequece i which the betwee successive terms is costat.. The sequece a a d is a arithmetic sequece i which a is the first term ad d is the. So for the arithmetic sequece a 5 the first term is, ad the commo differece is. 3. True or false? The th partial sum of a arithmetic sequece is the average of the first ad last terms times. 4. True or false? If we kow the first ad secod terms of a arithmetic sequece, the we ca fid ay other term. SKILLS 5 8 A sequece is give. (a) Fid the first five terms of the sequece. (b) What is the commo differece d? (c) Graph the terms you foud i (a). 5. a 5 6. a a 8. a 5 9 Fid the th term of the arithmetic sequece with give first term a ad commo differece d. What is the 0th term? 9. a 3, d 5 0. a 6, d 3. a 5, d. a 3, d Determie whether the sequece is arithmetic. If it is arithmetic, fid the commo differece. 3. 5,8,,4, ,6,9,3,... 5.,4,8,6,... 6.,4,6,8, , 3, 0,, l,l 4,l 8,l 6, , 4.3, 6.0, 7.7,... 0., 3, 4, 5,... 6 Fid the first five terms of the sequece, ad determie whether it is arithmetic. If it is arithmetic, fid the commo differece, ad express the th term of the sequece i the stadard form a a d.. a 4 7. a 4 3. a 4. a 5. a a 3 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

17 SECTION. Arithmetic Sequeces Determie the commo differece, the fifth term, the th term, ad the 00th term of the arithmetic sequece. 7.,5,8,,... 8.,5,9,3, ,9,4,9, ,8,5,,... 3., 8, 4,0, , 6.5, 8, 9.5, ,.3, 9.6, 6.9, , s, s, 3s, t, t 3, t 6, t 9, The teth term of a arithmetic sequece is, ad the secod 7 term is. Fid the first term. 38. The th term of a arithmetic sequece is 3, ad the fifth term is 8. Fid the 0th term. 39. The 00th term of a arithmetic sequece is 98, ad the commo differece is. Fid the first three terms. 40. The 0th term of a arithmetic sequece is 0, ad the commo differece is 3. Fid a formula for the th term. 4. Which term of the arithmetic sequece, 4, 7,... is 88? 4. The first term of a arithmetic sequece is, ad the commo differece is 4. Is,937 a term of this sequece? If so, which term is it? Fid the partial sum S of the arithmetic sequece that satisfies the give coditios. 43. a, d, a 3, d, 45. a 4, d, a 00, d 5, a 55, d, a 8, a 5 9.5, A partial sum of a arithmetic sequece is give. Fid the sum A B a 3 0.5k 54. k0 55. Show that a right triagle whose sides are i arithmetic progressio is similar to a triagle. 56. Fid the product of the umbers 57. A sequece is harmoic if the reciprocals of the terms of the sequece form a arithmetic sequece. Determie whether the followig sequece is harmoic:, 3 5, 3 7, 3, , 5 3, 3 6, 8 3,... 0 /0, 0 /0, 0 3/0, 0 4/0,..., 0 9/0 58. The harmoic mea of two umbers is the reciprocal of the average of the reciprocals of the two umbers. Fid the harmoic mea of 3 ad A arithmetic sequece has first term a 5 ad commo differece d. How may terms of this sequece must be added to get 700? 0 a A arithmetic sequece has first term a ad fourth term a 4 6. How may terms of this sequece must be added to get 356? APPLICATIONS 6. Depreciatio The purchase value of a office computer is $,500. Its aual depreciatio is $875. Fid the value of the computer after 6 years. 6. Poles i a Pile Telephoe poles are beig stored i a pile with 5 poles i the first layer, 4 i the secod, ad so o. If there are layers, how may telephoe poles does the pile cotai? 63. Salary Icreases A ma gets a job with a salary of $30,000 a year. He is promised a $300 raise each subsequet year. Fid his total earigs for a 0-year period. 64. Drive-I Theater A drive-i theater has spaces for 0 cars i the first parkig row, i the secod, 4 i the third, ad so o. If there are rows i the theater, fid the umber of cars that ca be parked. 65. Theater Seatig A architect desigs a theater with 5 seats i the first row, 8 i the secod, i the third, ad so o. If the theater is to have a seatig capacity of 870, how may rows must the architect use i his desig? 66. Fallig Ball Whe a object is allowed to fall freely ear the surface of the earth, the gravitatioal pull is such that the object falls 6 ft i the first secod, 48 ft i the ext secod, 80 ft i the ext secod, ad so o. (a) Fid the total distace a ball falls i 6 s. (b) Fid a formula for the total distace a ball falls i secods. 67. The Twelve Days of Christmas I the well-kow sog The Twelve Days of Christmas, a perso gives his sweetheart k gifts o the kth day for each of the days of Christmas. The perso also repeats each gift idetically o each subsequet day. Thus, o the th day the sweetheart receives a gift for the first day, gifts for the secod, 3 gifts for the third, ad so o. Show that the umber of gifts received o the th day is a partial sum of a arithmetic sequece. Fid this sum. DISCOVERY DISCUSSION WRITING 68. Arithmetic Meas The arithmetic mea (or average) of two umbers a ad b is m a b Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

18 800 CHAPTER Sequeces ad Series Note that m is the same distace from a as from b, so a, m, b is a arithmetic sequece. I geeral, if m, m,...,m k are equally spaced betwee a ad b so that a, m, m,..., m k, b is a arithmetic sequece, the m, m,...,m k are called k arithmetic meas betwee a ad b. (a) Isert two arithmetic meas betwee 0 ad 8. (b) Isert three arithmetic meas betwee 0 ad 8. (c) Suppose a doctor eeds to icrease a patiet s dosage of a certai medicie from 00 mg to 300 mg per day i five equal steps. How may arithmetic meas must be iserted betwee 00 ad 300 to give the progressio of daily doses, ad what are these meas?.3 GEOMETRIC SEQUENCES Geometric Sequeces Partial Sums of Geometric Sequeces What Is a Ifiite Series? Ifiite Geometric Series I this sectio we study geometric sequeces. This type of sequece occurs frequetly i applicatios to fiace, populatio growth, ad other fields. Geometric Sequeces Recall that a arithmetic sequece is geerated whe we repeatedly add a umber d to a iitial term a. A geometric sequece is geerated whe we start with a umber a ad repeatedly multiply by a fixed ozero costat r. DEFINITION OF A GEOMETRIC SEQUENCE A geometric sequece is a sequece of the form a, ar, ar, ar 3, ar 4,... The umber a is the first term, ad r is the commo ratio of the sequece. The th term of a geometric sequece is give by a ar The umber r is called the commo ratio because the ratio of ay two cosecutive terms of the sequece is r. EXAMPLE Geometric Sequeces (a) If a 3 ad r, the we have the geometric sequece or 3, 3 #, 3 #, 3 # 3, 3 # 4,... 3, 6,, 4, 48,... Notice that the ratio of ay two cosecutive terms is r. The th term is a 3. (b) The sequece, 0, 50, 50, 50,... is a geometric sequece with a ad r 5. Whe r is egative, the terms of the sequece alterate i sig. The th term is a 5. (c) The sequece, 3, 9, 7, 8,... is a geometric sequece with a ad r. The th term is a A 3B 3. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

19 SECTION.3 Geometric Sequeces FIGURE m 3 9 m m h 0 3 t FIGURE (d) The graph of the geometric sequece a # 5 is show i Figure. Notice that the poits i the graph lie o the graph of the expoetial fuctio y # 5 x. If 0 r, the the terms of the geometric sequece ar decrease, but if r, the the terms icrease. (What happes if r?) NOW TRY EXERCISES 5, 9, AND3 Geometric sequeces occur aturally. Here is a simple example. Suppose a ball has elasticity such that whe it is dropped, it bouces up oe-third of the distace it has falle. If this ball is dropped from a height of m, the it bouces up to a height of A 3B. O its secod bouce, it returs to a height of A 3BA 3B 3 m 9 m, ad so o (see Figure ). Thus, the height h that the ball reaches o its th bouce is give by the geometric sequece We ca fid the th term of a geometric sequece if we kow ay two terms, as the followig examples show. EXAMPLE h 3A 3B A 3B Fidig Terms of a Geometric Sequece Fid the eighth term of the geometric sequece 5, 5, 45,... SOLUTION To fid a formula for the th term of this sequece, we eed to fid a ad r. Clearly, a 5. To fid r, we fid the ratio of ay two cosecutive terms. For istace, r Thus a 53 The eighth term is a ,935. NOW TRY EXERCISE 7 EXAMPLE 3 Fidig Terms of a Geometric Sequece The third term of a geometric sequece is, ad the sixth term is. Fid the fifth term. SOLUTION Sice this sequece is geometric, its th term is give by the formula a ar. Thus From the values we are give for these two terms, we get the followig system of equatios: We solve this system by dividig. a 3 ar 3 ar a 6 ar 6 5 ar u ar 70 3 ar ar 5 70 ar 3 r r Simplify Take cube root of each side Substitutig for r i the first equatio, 4 ar, gives 63 4 aa 3 B a 7 63 Solve for a Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

20 80 CHAPTER Sequeces ad Series The Grager Collectio, New York. All rights reserved SRINIVASA RAMANUJAN (887 90) was bor ito a poor family i the small tow of Kumbakoam i Idia. Self-taught i mathematics, he worked i virtual isolatio from other mathematicias. At the age of 5 he wrote a letter to G. H. Hardy, the leadig British mathematicia at the time, listig some of his discoveries. Hardy immediately recogized Ramauja s geius, ad for the ext six years the two worked together i Lodo util Ramauja fell ill ad retured to his hometow i Idia, where he died a year later. Ramauja was a geius with pheomeal ability to see hidde patters i the properties of umbers. Most of his discoveries were writte as complicated ifiite series, the importace of which was ot recogized util may years after his death. I the last year of his life he wrote 30 pages of mysterious formulas, may of which still defy proof. Hardy tells the story that whe he visited Ramauja i a hospital ad arrived i a taxi, he remarked to Ramauja that the cab s umber, 79, was uiterestig. Ramauja replied No, it is a very iterestig umber. It is the smallest umber expressible as the sum of two cubes i two differet ways. It follows that the th term of this sequece is Thus the fifth term is NOW TRY EXERCISE 37 Partial Sums of Geometric Sequeces For the geometric sequece a, ar, ar, ar 3, ar 4,...,ar,...,theth partial sum is To fid a formula for S, we multiply S by r ad subtract from S. So We summarize this result. EXAMPLE 4 S a k rs S rs a ar Fidig a Partial Sum of a Geometric Sequece Fid the sum of the first five terms of the geometric sequece SOLUTION The required sum is the sum of the first five terms of a geometric sequece with a ad r 0.7. Usig the formula for S with 5, we get Thus the sum of the first five terms of this sequece is.773. NOW TRY EXERCISES 43 AND 47 a 7A 3 B a 5 7A 3 B 5 7A 3 B ar k a ar ar ar 3 ar 4... ar S a ar ar ar 3 ar 4... ar ar ar ar 3 ar 4... ar ar S r a r S a r r PARTIAL SUMS OF A GEOMETRIC SEQUENCE, 0.7, 0.49, 0.343,... r For the geometric sequece a ar, the th partial sum is give by S a ar ar ar 3 ar 4... ar r S a r r 0.75 S 5 # EXAMPLE 5 Fidig a Partial Sum of a Geometric Sequece Fid the sum a 7A 3B k. 5 k Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

21 SECTION.3 Geometric Sequeces 803 SOLUTION The give sum is the fifth partial sum of a geometric sequece with first term a 7A 3B 4 3 ad commo ratio r 3. Thus, by the formula for S we have NOW TRY EXERCISE 49 4 A S 5 # 3B 5 3 A 3B 4 3 # What Is a Ifiite Series? A expressio of the form q a a k a a a 3 a 4... k is called a ifiite series. The dots mea that we are to cotiue the additio idefiitely. What meaig ca we attach to the sum of ifiitely may umbers? It seems at first that it is ot possible to add ifiitely may umbers ad arrive at a fiite umber. But cosider the followig problem. You have a cake, ad you wat to eat it by first eatig half the cake, the eatig half of what remais, the agai eatig half of what remais. This process ca cotiue idefiitely because at each stage, some of the cake remais. (See Figure 3.) FIGURE 3 Does this mea that it s impossible to eat all of the cake? Of course ot. Let s write dow what you have eate from this cake: q a k k This is a ifiite series, ad we ote two thigs about it: First, from Figure 3 it s clear that o matter how may terms of this series we add, the total will ever exceed. Secod, the more terms of this series we add, the closer the sum is to (see Figure 3). This suggests that the umber ca be writte as the sum of ifiitely may smaller umbers: To make this more precise, let s look at the partial sums of this series: S S S S ad, i geeral (see Example 5 of Sectio.), S Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

22 804 CHAPTER Sequeces ad Series As gets larger ad larger, we are addig more ad more of the terms of this series. Ituitively, as gets larger, S gets closer to the sum of the series. Now otice that as gets large, / gets closer ad closer to 0. Thus S gets close to 0. Usig the otatio of Sectio 3.7, we ca write S as q I geeral, if S gets close to a fiite umber S as gets large, we say that the ifiite series coverges (or is coverget). The umber S is called the sum of the ifiite series. If a ifiite series does ot coverge, we say that the series diverges (or is diverget). Here is aother way to arrive at the formula for the sum of a ifiite geometric series: S a ar ar ar 3... a r a ar ar... a rs Solve the equatio S a rs for S to get S rs a rs a S a r Ifiite Geometric Series A ifiite geometric series is a series of the form a ar ar ar 3 ar 4... ar... We ca apply the reasoig used earlier to fid the sum of a ifiite geometric series. The th partial sum of such a series is give by the formula r S a r r It ca be show that if 0 r 0, the r gets close to 0 as gets large (you ca easily covice yourself of this usig a calculator). It follows that S gets close to a/ r as gets large, or S a as q r Thus the sum of this ifiite geometric series is a/ r. SUM OF AN INFINITE GEOMETRIC SERIES If 0 r 0, the the ifiite geometric series coverges ad has the sum If q 0 r 0, the series diverges. a ar k a ar ar ar 3... k S a r MATHEMATICS IN THE MODERN WORLD Bill Ross/CORBIS Fractals May of the thigs we model i this book have regular predictable shapes. But recet advaces i mathematics have made it possible to model such seemigly radom or eve chaotic shapes as those of a cloud, a flickerig flame, a moutai, or a jagged coastlie. The basic tools i this type of modelig are the fractals iveted by the mathematicia Beoit Madelbrot. A fractal is a geometric shape built up from a simple basic shape by scalig ad repeatig the shape idefiitely accordig to a give rule. Fractals have ifiite detail; this meas the closer you look, the more you see. They are also self-similar; that is, zoomig i o a portio of the fractal yields the same detail as the origial shape. Because of their beautiful shapes, fractals are used by movie makers to create fictioal ladscapes ad exotic backgrouds. Although a fractal is a complex shape, it is produced accordig to very simple rules. This property of fractals is exploited i a process of storig pictures o a computer called fractal image compressio. I this process a picture is stored as a simple basic shape ad a rule; repeatig the shape accordig to the rule produces the origial picture. This is a extremely efficiet method of storage; that s how thousads of color pictures ca be put o a sigle compact disc. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

23 EXAMPLE 6 Ifiite Series Determie whether the ifiite geometric series is coverget or diverget. If it is coverget, fid its sum. (a) (b) SOLUTION (a) This is a ifiite geometric series with a ad r 5. Sice 0 r the series coverges. By the formula for the sum of a ifiite geometric series we have S 5 5 SECTION.3 Geometric Sequeces a 7 5 b a b... (b) This is a ifiite geometric series with a ad r 7 5. Sice 0 r the series diverges. 7 NOW TRY EXERCISES 5 AND 55 EXAMPLE 7 Writig a Repeated Decimal as a Fractio Fid the fractio that represets the ratioal umber.35. SOLUTION This repeatig decimal ca be writte as a series: , ,000,000 5,000,000, After the first term, the terms of this series form a ifiite geometric series with Thus the sum of this part of the series is So a ad r S 99 5 # NOW TRY EXERCISE 63.3 EXERCISES CONCEPTS. A geometric sequece is a sequece i which the of successive terms is costat.. The sequece a ar is a geometric sequece i which a is the first term ad r is the. So for the geometric sequece a 5 the first term is, ad the commo ratio is. 3. True or false? If we kow the first ad secod terms of a geometric sequece, the we ca fid ay other term. 4. (a) The th partial sum of a geometric sequece a ar is give by S. (b) The series q a ar k a ar ar ar 3... k is a ifiite series. If 0 r 0, the this series, ad its sum is S. If 0 r 0, the series. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

24 806 CHAPTER Sequeces ad Series SKILLS 5 8 The th term of a sequece is give. (a) Fid the first five terms of the sequece. (b) What is the commo ratio r? (c) Graph the terms you foud i (a). 5. a a 8. a 3 5 A 9 Fid the th term of the geometric sequece with give first term a ad commo ratio r. What is the fourth term? 9. a 3, r 5 0. a 6, r 3. a 5, r. a 3, r Determie whether the sequece is geometric. If it is geometric, fid the commo ratio. 3.,4,8,6,... 4.,6,8,36, , 3, 3 4, 3 8, , 9, 3,,... 7., 3, 4, 5, e, e 4, e 6, e 8, ,.,.,.33,... 0., 4, 6, 8,... 6 Fid the first five terms of the sequece, ad determie whether it is geometric. If it is geometric, fid the commo ratio, ad express the th term of the sequece i the stadard form a ar.. a. a a 4. a 4 5. a 6. a l Determie the commo ratio, the fifth term, ad the th term of the geometric sequece. 7.,6,8,54, , 4 3, 8 9, 56 7, , 0.09, 0.07, 0.008, ,,,, ,,,, ,,, 8, , 3 5/3,3 7/3,7, t, t, t 3 4, t 4 8, , s /7, s 4/ 7, s 6/7, , 5 c,5 c,5 3c, The first term of a geometric sequece is 8, ad the secod term is 4. Fid the fifth term. 38. The first term of a geometric sequece is 3, ad the third term 4 is. Fid the fifth term The commo ratio i a geometric sequece is 5, ad the fourth 5 term is. Fid the third term. B a The commo ratio i a geometric sequece is, ad the fifth term is. Fid the first three terms. 4. Which term of the geometric sequece, 6, 8,... is 8,098? 4. The secod ad fifth terms of a geometric sequece are 0 ad 50, respectively. Is 3,50 a term of this sequece? If so, which term is it? Fid the partial sum S of the geometric sequece that satisfies the give coditios. 43. a 5, r, a 3, r 3, a 3 8, a 6 4, a 0., a , Fid the sum a 3A B k Determie whether the ifiite geometric series is coverget or diverget. If it is coverget, fid its sum a 3 b a 3 3 b k Express the repeatig decimal as a fractio If the umbers a, a,...,a form a geometric sequece, the a, a 3,...,a are geometric meas betwee a ad a. Isert three geometric meas betwee 5 ad Fid the sum of the first te terms of the sequece APPLICATIONS a b, a b, a 3 3b, a 4 4b, Depreciatio A costructio compay purchases a bulldozer for $60,000. Each year the value of the bulldozer depreciates by 0% of its value i the precedig year. Let V be the value of the bulldozer i the th year. (Let be the year the bulldozer is purchased.) (a) Fid a formula for V. (b) I what year will the value of the bulldozer be less tha $00,000? 5 a j0 7A 3 j B Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

25 SECTION.3 Geometric Sequeces Family Tree A perso has two parets, four gradparets, eight great-gradparets, ad so o. How may acestors does a perso have 5 geeratios back? Father Mother Gradfather Gradmother Gradfather Gradmother 73. Boucig Ball A ball is dropped from a height of 80 ft. The elasticity of this ball is such that it rebouds three-fourths of the distace it has falle. How high does the ball reboud o the fifth bouce? Fid a formula for how high the ball rebouds o the th bouce. 74. Bacteria Culture A culture iitially has 5000 bacteria, ad its size icreases by 8% every hour. How may bacteria are preset at the ed of 5 hours? Fid a formula for the umber of bacteria preset after hours. 75. Mixig Coolat A truck radiator holds 5 gal ad is filled with water. A gallo of water is removed from the radiator ad replaced with a gallo of atifreeze; the a gallo of the mixture is removed from the radiator ad agai replaced by a gallo of atifreeze. This process is repeated idefiitely. How much water remais i the tak after this process is repeated 3 times? 5 times? times? 76. Musical Frequecies The frequecies of musical otes (measured i cycles per secod) form a geometric sequece. Middle C has a frequecy of 56, ad the C that is a octave higher has a frequecy of 5. Fid the frequecy of C two octaves below middle C. 80. Drug Cocetratio A certai drug is admiistered oce a day. The cocetratio of the drug i the patiet s bloodstream icreases rapidly at first, but each successive dose has less effect tha the precedig oe. The total amout of the drug (i mg) i the bloodstream after the th dose is give by (a) Fid the amout of the drug i the bloodstream after 0 days. (b) If the drug is take o a log-term basis, the amout i the bloodstream is approximated by the ifiite series q a 50A B k. Fid the sum of this series. k a 50A B k k 8. Boucig Ball A certai ball rebouds to half the height from which it is dropped. Use a ifiite geometric series to approximate the total distace the ball travels after beig dropped from m above the groud util it comes to rest. 8. Boucig Ball If the ball i Exercise 8 is dropped from a height of 8 ft, the s is required for its first complete bouce from the istat it first touches the groud util it ext touches the groud. Each subsequet complete bouce requires / as log as the precedig complete bouce. Use a ifiite geometric series to estimate the time iterval from the istat the ball first touches the groud util it stops boucig. 83. Geometry The midpoits of the sides of a square of side are joied to form a ew square. This procedure is repeated for each ew square. (See the figure.) (a) Fid the sum of the areas of all the squares. (b) Fid the sum of the perimeters of all the squares. 77. Boucig Ball A ball is dropped from a height of 9 ft. The elasticity of the ball is such that it always bouces up oethird the distace it has falle. (a) Fid the total distace the ball has traveled at the istat it hits the groud the fifth time. (b) Fid a formula for the total distace the ball has traveled at the istat it hits the groud the th time. 78. Geometric Savigs Pla A very patiet woma wishes to become a billioaire. She decides to follow a simple scheme: She puts aside cet the first day, cets the secod day, 4 cets the third day, ad so o, doublig the umber of cets each day. How much moey will she have at the ed of 30 days? How may days will it take this woma to realize her wish? 79. St. Ives The followig is a well-kow childre s rhyme: As I was goig to St. Ives, I met a ma with seve wives; Every wife had seve sacks; Every sack had seve cats; Every cat had seve kits; Kits, cats, sacks, ad wives, How may were goig to St. Ives? Assumig that the etire group is actually goig to St. Ives, show that the aswer to the questio i the rhyme is a partial sum of a geometric sequece, ad fid the sum. 84. Geometry A circular disk of radius R is cut out of paper, as show i figure (a). Two disks of radius R are cut out of paper ad placed o top of the first disk, as i figure (b), ad the four disks of radius 4 R are placed o these two disks, as i figure (c). Assumig that this process ca be repeated idefiitely, fid the total area of all the disks. (a) (b) (c) Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

26 808 CHAPTER Sequeces ad Series 85. Geometry A yellow square of side is divided ito ie smaller squares, ad the middle square is colored blue as show i the figure. Each of the smaller yellow squares is i tur divided ito ie squares, ad each middle square is colored blue. If this process is cotiued idefiitely, what is the total area that is colored blue? DISCOVERY DISCUSSION WRITING 86. Arithmetic or Geometric? The first four terms of a sequece are give. Determie whether these terms ca be the terms of a arithmetic sequece, a geometric sequece, or either. Fid the ext term if the sequece is arithmetic or geometric. 5 7 (a) 5, 3, 5, 3,... (b) 3,, 3, 3,... (c) 3,3,33,9,... (d),,,,... (e),,,,... (f) x, x, x, x, (g) 3,,0,,... (h) 5, 3 5, 6 5,, Reciprocals of a Geometric Sequece If a, a, a 3,... is a geometric sequece with commo ratio r, show that the sequece a, a, a 3,... is also a geometric sequece, ad fid the commo ratio. 88. Logarithms of a Geometric Sequece If a, a, a 3,... is a geometric sequece with a commo ratio r 0 ad a 0, show that the sequece log a, log a, log a 3,... is a arithmetic sequece, ad fid the commo differece. 89. Expoetials of a Arithmetic Sequece If a, a, a 3,... is a arithmetic sequece with commo differece d, show that the sequece 0 a, 0a, 0a 3,... is a geometric sequece, ad fid the commo ratio. DISCOVERY PROJECT Fidig Patters I this project we ivestigate the process of fidig patters i sequeces by usig differece sequeces. You ca fid the project at the book compaio website: MATHEMATICS OF FINANCE The Amout of a Auity The Preset Value of a Auity Istallmet Buyig May fiacial trasactios ivolve paymets that are made at regular itervals. For example, if you deposit $00 each moth i a iterest-bearig accout, what will the value of your accout be at the ed of 5 years? If you borrow $00,000 to buy a house, how much must your mothly paymets be i order to pay off the loa i 30 years? Each of these questios ivolves the sum of a sequece of umbers; we use the results of the precedig sectio to aswer them here. The Amout of a Auity A auity is a sum of moey that is paid i regular equal paymets. Although the word auity suggests aual (or yearly) paymets, they ca be made semiaually, quarterly, mothly, or at some other regular iterval. Paymets are usually made at the ed of the paymet iterval. The amout of a auity is the sum of all the idividual paymets from the time of the first paymet util the last paymet is made, together with all the iterest. We deote this sum by A f (the subscript f here is used to deote fial amout). EXAMPLE Calculatig the Amout of a Auity A ivestor deposits $400 every December 5 ad Jue 5 for 0 years i a accout that ears iterest at the rate of 8% per year, compouded semiaually. How much will be i the accout immediately after the last paymet? Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

27 SECTION.4 Mathematics of Fiace 809 Whe usig iterest rates i calculators, remember to covert percetages to decimals. For example, 8% is SOLUTION We eed to fid the amout of a auity cosistig of 0 semiaual paymets of $400 each. Sice the iterest rate is 8% per year, compouded semiaually, the iterest rate per time period is i 0.08/ The first paymet is i the accout for 9 time periods, the secod for 8 time periods, ad so o. The last paymet receives o iterest. The situatio ca be illustrated by the time lie i Figure. Time (years) NOW Paymet (dollars) (.04) 400(.04) 400(.04) 3 FIGURE 400(.04) 4 400(.04) 5 400(.04) 6 400(.04) 7 400(.04) 8 400(.04) 9 The amout A f of the auity is the sum of these 0 amouts. Thus A f But this is a geometric series with a 400, r.04, ad 0, so.04 0 A f 400, Thus the amout i the accout after the last paymet is $,9.3. NOW TRY EXERCISE 3 I geeral, the regular auity paymet is called the periodic ret ad is deoted by R. We also let i deote the iterest rate per time period ad let deote the umber of paymets. We always assume that the time period i which iterest is compouded is equal to the time betwee paymets. By the same reasoig as i Example, we see that the amout A f of a auity is A f R R i R i... R i Sice this is the th partial sum of a geometric sequece with a R ad r i, the formula for the partial sum gives i i A f R R R i i i i AMOUNT OF AN ANNUITY The amout A f of a auity cosistig of regular equal paymets of size R with iterest rate i per time period is give by i A f R i Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

28 80 CHAPTER Sequeces ad Series MATHEMATICS IN THE MODERN WORLD Mathematical Ecoomics The health of the global ecoomy is determied by such iterrelated factors as supply, demad, productio, cosumptio, pricig, distributio, ad thousads of other factors. These factors are i tur determied by ecoomic decisios (for example, whether or ot you buy a certai brad of toothpaste) made by billios of differet idividuals each day. How will today s creatio ad distributio of goods affect tomorrow s ecoomy? Such questios are tackled by mathematicias who work o mathematical models of the ecoomy. I the 940s Wassily Leotief, a pioeer i this area, created a model cosistig of thousads of equatios that describe how differet sectors of the ecoomy, such as the oil idustry, trasportatio, ad commuicatio, iteract with each other. A differet approach to ecoomic models, oe dealig with idividuals i the ecoomy as opposed to large sectors, was pioeered by Joh Nash i the 950s. I his model, which uses game theory, the ecoomy is a game where idividual players make decisios that ofte lead to mutual gai. Leotief ad Nash were awarded the Nobel Prize i Ecoomics i 973 ad 994, respectively. Ecoomic theory cotiues to be a major area of mathematical research. EXAMPLE Calculatig the Amout of a Auity How much moey should be ivested every moth at % per year, compouded mothly, i order to have $4000 i 8 moths? SOLUTION I this problem i 0./ 0.0, A f 4000, ad 8. We eed to fid the amout R of each paymet. By the formula for the amout of a auity, Solvig for R, we get Thus the mothly ivestmet should be $ NOW TRY EXERCISE 9 The Preset Value of a Auity If you were to receive $0,000 five years from ow, it would be worth much less tha if you got $0,000 right ow. This is because of the iterest you could accumulate durig the ext five years if you ivested the moey ow. What smaller amout would you be willig to accept ow istead of receivig $0,000 i five years? This is the amout of moey that, together with iterest, would be worth $0,000 i five years. The amout that we are lookig for here is called the discouted value or preset value. If the iterest rate is 8% per year, compouded quarterly, the the iterest per time period is i 0.08/4 0.0, ad there are time periods. If we let PV deote the preset value, the by the formula for compoud iterest (Sectio 4.), we have so 4000 R R ,000 PV i PV PV 0, Thus i this situatio the preset value of $0,000 is $ This reasoig leads to a geeral formula for preset value. If a amout A f is to be paid i a lump sum time periods from ow ad the iterest rate per time period is i, the its preset value A p is give by A p A f i Similarly, the preset value of a auity is the amout A p that must be ivested ow at the iterest rate i per time period to provide paymets, each of amout R. Clearly, A p is the sum of the preset values of each idividual paymet (see Exercise 9). Aother way of fidig A p is to ote that A p is the preset value of A f : i A p A f i R i i R i i THE PRESENT VALUE OF AN ANNUITY The preset value A p of a auity cosistig of regular equal paymets of size R ad iterest rate i per time period is give by i A p R i Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

29 EXAMPLE 3 Calculatig the Preset Value of a Auity A perso wis $0,000,000 i the Califoria lottery, ad the amout is paid i yearly istallmets of half a millio dollars each for 0 years. What is the preset value of his wiigs? Assume that he ca ear 0% iterest, compouded aually. SOLUTION Sice the amout wo is paid as a auity, we eed to fid its preset value. Here, i 0., R $500,000, ad 0. Thus This meas that the wier really wo oly $4,56,78.86 if it were paid immediately. NOW TRY EXERCISE SECTION.4 Mathematics of Fiace A p 500,000 4,56, Istallmet Buyig Whe you buy a house or a car by istallmet, the paymets that you make are a auity whose preset value is the amout of the loa. EXAMPLE 4 The Amout of a Loa A studet wishes to buy a car. She ca afford to pay $00 per moth but has o moey for a dow paymet. If she ca make these paymets for four years ad the iterest rate is %, what purchase price ca she afford? SOLUTION The paymets that the studet makes costitute a auity whose preset value is the price of the car (which is also the amout of the loa, i this case). Here, we have i 0./ 0.0, R 00, ad 4 48, so i A p R i 0.0 Thus the studet ca buy a car priced at $ NOW TRY EXERCISE 9 Whe a bak makes a loa that is to be repaid with regular equal paymets R, the the paymets form a auity whose preset value A p is the amout of the loa. So to fid the size of the paymets, we solve for R i the formula for the amout of a auity. This gives the followig formula for R. INSTALLMENT BUYING If a loa A p is to be repaid i regular equal paymets with iterest rate i per time period, the the size R of each paymet is give by ia p R i EXAMPLE 5 Calculatig Mothly Mortgage Paymets A couple borrows $00,000 at 9% iterest as a mortage loa o a house. They expect to make mothly paymets for 30 years to repay the loa. What is the size of each paymet? Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

30 8 CHAPTER Sequeces ad Series SOLUTION The mortgage paymets form a auity whose preset value is A p $00,000. Also, i 0.09/ , ad We are lookig for the amout R of each paymet. From the formula for istallmet buyig, we get ia p ,000 R i Thus the mothly paymets are $ NOW TRY EXERCISE 5 We ow illustrate the use of graphig devices i solvig problems related to istallmet buyig. EXAMPLE 6 Calculatig the Iterest Rate from the Size of Mothly Paymets A car dealer sells a ew car for $8,000. He offers the buyer paymets of $405 per moth for 5 years. What iterest rate is this car dealer chargig? SOLUTION The paymets form a auity with preset value A p $8,000, R 405, ad To fid the iterest rate, we must solve for i i the equatio FIGURE A little experimetatio will covice you that it is ot possible to solve this equatio for i algebraically. So to fid i, we use a graphig device to graph R as a fuctio of the iterest rate x, ad we the use the graph to fid the iterest rate correspodig to the value of R we wat ($405 i this case). Sice i x/, we graph the fuctio i the viewig rectagle 30.06, , 4504, as show i Figure. We also graph the horizotal lie Rx 405 i the same viewig rectagle. The, by movig the cursor to the poit of itersectio of the two graphs, we fid that the correspodig x-value is approximately 0.5. Thus the iterest rate is about %. NOW TRY EXERCISE 5 ia p R i Rx x 8,000 a x b 60.4 EXERCISES CONCEPTS. A auity is a sum of moey that is paid i regular equal paymets. The of a auity is the sum of all the idividual paymets together with all the iterest.. The of a auity is the amout that must be ivested ow at iterest rate i per time period to provide paymets each of amout R. APPLICATIONS 3. Auity Fid the amout of a auity that cosists of 0 aual paymets of $000 each ito a accout that pays 6% iterest per year. 4. Auity Fid the amout of a auity that cosists of 4 mothly paymets of $500 each ito a accout that pays 8% iterest per year, compouded mothly. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

31 SECTION.4 Mathematics of Fiace Auity Fid the amout of a auity that cosists of 0 aual paymets of $5000 each ito a accout that pays iterest of % per year. 6. Auity Fid the amout of a auity that cosists of 0 semiaual paymets of $500 each ito a accout that pays 6% iterest per year, compouded semiaually. 7. Auity Fid the amout of a auity that cosists of 6 quarterly paymets of $300 each ito a accout that pays 8% iterest per year, compouded quarterly. 8. Auity Fid the amout of a auity that cosists of 40 aual paymets of $000 each ito a accout that pays iterest of 5% per year. 9. Savig How much moey should be ivested every quarter at 0% per year, compouded quarterly, to have $5000 i years? 0. Savig How much moey should be ivested mothly at 6% per year, compouded mothly, to have $000 i 8 moths?. Auity What is the preset value of a auity that cosists of 0 semiaual paymets of $000 at a iterest rate of 9% per year, compouded semiaually?. Auity What is the preset value of a auity that cosists of 30 mothly paymets of $300 at a iterest rate of 8% per year, compouded mothly. 3. Fudig a Auity How much moey must be ivested ow at 9% per year, compouded semiaually, to fud a auity of 0 paymets of $00 each, paid every 6 moths, the first paymet beig 6 moths from ow? 4. Fudig a Auity A 55-year-old ma deposits $50,000 to fud a auity with a isurace compay. The moey will be ivested at 8% per year, compouded semiaually. He is to draw semiaual paymets util he reaches age 65. What is the amout of each paymet? 5. Fiacig a Car A woma wats to borrow $,000 to buy a car. She wats to repay the loa by mothly istallmets for 4 years. If the iterest rate o this loa is 0 % per year, compouded mothly, what is the amout of each paymet? 6. Mortgage What is the mothly paymet o a 30-year mortgage of $80,000 at 9% iterest? What is the mothly paymet o this same mortgage if it is to be repaid over a 5-year period? 7. Mortgage What is the mothly paymet o a 30-year mortgage of $00,000 at 8% iterest per year, compouded mothly? What is the total amout paid o this loa over the 30-year period? 8. Mortgage What is the mothly paymet o a 5-year mortgage of $00,000 at 6% iterest? What is the total amout paid o this loa over the 5-year period? 9. Mortgage Dr. Gupta is cosiderig a 30-year mortgage at 6% iterest. She ca make paymets of $3500 a moth. What size loa ca she afford? 0. Mortgage A couple ca afford to make a mothly mortgage paymet of $650. If the mortgage rate is 9% ad the cou- ple iteds to secure a 30-year mortgage, how much ca they borrow?. Fiacig a Car Jae agrees to buy a car for a dow paymet of $000 ad paymets of $0 per moth for 3 years. If the iterest rate is 8% per year, compouded mothly, what is the actual purchase price of her car?. Fiacig a Rig Mike buys a rig for his fiacee by payig $30 a moth for oe year. If the iterest rate is 0% per year, compouded mothly, what is the price of the rig? 3. Mortgage A couple secures a 30-year loa of $00,000 at 3 9 4% per year, compouded mothly, to buy a house. (a) What is the amout of their mothly paymet? (b) What total amout will they pay over the 30-year period? (c) If, istead of takig the loa, the couple deposits the 3 mothly paymets i a accout that pays 9 4% iterest per year, compouded mothly, how much will be i the accout at the ed of the 30-year period? 4. Mortgage A couple eeds a mortgage of $300,000. Their mortgage broker presets them with two optios: a 30-year 3 mortgage at 6 % iterest or a 5-year mortgage at 5 4% iterest. (a) Fid the mothly paymet o the 30-year mortgage ad o the 5-year mortgage. Which mortgage has the larger mothly paymet? (b) Fid the total amout to be paid over the life of each loa. Which mortgage has the lower total paymet over its lifetime? 5. Iterest Rate Joh buys a stereo system for $640. He agrees to pay $3 a moth for years. Assumig that iterest is compouded mothly, what iterest rate is he payig? 6. Iterest Rate Jaet s paymets o her $,500 car are $40 a moth for 3 years. Assumig that iterest is compouded mothly, what iterest rate is she payig o the car loa? 7. Iterest Rate A item at a departmet store is priced at $89.99 ad ca be bought by makig 0 paymets of $0.50. Fid the iterest rate, assumig that iterest is compouded mothly. 8. Iterest Rate A ma purchases a $000 diamod rig for a dow paymet of $00 ad mothly istallmets of $88 for years. Assumig that iterest is compouded mothly, what iterest rate is he payig? DISCOVERY DISCUSSION WRITING 9. Preset Value of a Auity (a) Draw a time lie as i Example to show that the preset value of a auity is the sum of the preset values of each paymet, that is, A p R i R i R i... R 3 i (b) Use part (a) to derive the formula for A p giveithetext. 30. A Auity That Lasts Forever A auity i perpetuity is oe that cotiues forever. Such auities are useful i settig up scholarship fuds to esure that the award cotiues. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

32 84 CHAPTER Sequeces ad Series (a) Draw a time lie (as i Example ) to show that to set up a auity i perpetuity of amout R per time period, the amout that must be ivested ow is A p R i R i R i... R 3 i... where i is the iterest rate per time period. (b) Fid the sum of the ifiite series i part (a) to show that A p R i (c) How much moey must be ivested ow at 0% per year, compouded aually, to provide a auity i perpetuity of $5000 per year? The first paymet is due i oe year. (d) How much moey must be ivested ow at 8% per year, compouded quarterly, to provide a auity i perpetuity of $3000 per year? The first paymet is due i oe year. 3. Amortizig a Mortgage Whe they bought their house, Joh ad Mary took out a $90,000 mortgage at 9% iterest, repayable mothly over 30 years. Their paymet is $74.7 per moth (check this, usig the formula i the text). The bak gave them a amortizatio schedule, which is a table showig how much of each paymet is iterest, how much goes to- ward the pricipal, ad the remaiig pricipal after each paymet. The table below shows the first few etries i the amortizatio schedule. Paymet Total Iterest Pricipal Remaiig umber paymet paymet paymet pricipal , , , ,80.0 After 0 years they have made 0 paymets ad are woderig how much they still owe, but they have lost the amortizatio schedule. (a) How much do Joh ad Mary still owe o their mortgage? [Hit: The remaiig balace is the preset value of the 40 remaiig paymets.] (b) How much of their ext paymet is iterest, ad how much goes toward the pricipal? [Hit: Sice 9% 0.75%, they must pay 0.75% of the remaiig pricipal i iterest each moth.].5 MATHEMATICAL INDUCTION Cojecture ad Proof Mathematical Iductio There are two aspects to mathematics discovery ad proof ad they are of equal importace. We must discover somethig before we ca attempt to prove it, ad we caot be certai of its truth util it has bee proved. I this sectio we examie the relatioship betwee these two key compoets of mathematics more closely. Cojecture ad Proof Let s try a simple experimet. We add more ad more of the odd umbers as follows: What do you otice about the umbers o the right side of these equatios? They are, i fact, all perfect squares. These equatios say the followig: The sum of the first odd umber is. The sum of the first odd umbers is. The sum of the first 3 odd umbers is 3. The sum of the first 4 odd umbers is 4. The sum of the first 5 odd umbers is 5. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

33 SECTION.5 Mathematical Iductio 85 Cosider the polyomial p 4 Here are some values of p: p 4 p 43 p3 47 p4 53 p5 6 p6 7 p7 83 p8 97 All the values so far are prime umbers. I fact, if you keep goig, you will fid that p is prime for all atural umbers up to 40. It might seem reasoable at this poit to cojecture that p is prime for every atural umber. But that cojecture would be too hasty, because it is easily see that p4 is ot prime. This illustrates that we caot be certai of the truth of a statemet o matter how may special cases we check. We eed a covicig argumet a proof to determie the truth of a statemet. This leads aturally to the followig questio: Is it true that for every atural umber, the sum of the first odd umbers is? Could this remarkable property be true? We could try a few more umbers ad fid that the patter persists for the first 6, 7, 8, 9, ad 0 odd umbers. At this poit we feel quite sure that this is always true, so we make a cojecture: The sum of the first odd umbers is. Sice we kow that the th odd umber is, we ca write this statemet more precisely as It is importat to realize that this is still a cojecture. We caot coclude by checkig a fiite umber of cases that a property is true for all umbers (there are ifiitely may). To see this more clearly, suppose someoe tells us that he has added up the first trillio odd umbers ad foud that they do ot add up to trillio squared. What would you tell this perso? It would be silly to say that you re sure it s true because you have already checked the first five cases. You could, however, take out paper ad pecil ad start checkig it yourself, but this task would probably take the rest of your life. The tragedy would be that after completig this task, you would still ot be sure of the truth of the cojecture! Do you see why? Herei lies the power of mathematical proof. A proof is a clear argumet that demostrates the truth of a statemet beyod doubt. Mathematical Iductio Let s cosider a special kid of proof called mathematical iductio. Here is how it works: Suppose we have a statemet that says somethig about all atural umbers. For example, for ay atural umber, let P be the followig statemet: P: The sum of the first odd umbers is Sice this statemet is about all atural umbers, it cotais ifiitely may statemets; we will call them P(), P(),... P: The sum of the first odd umber is. P: The sum of the first odd umbers is. P3: The sum of the first 3 odd umbers is 3... How ca we prove all of these statemets at oce? Mathematical iductio is a clever way of doig just that. The crux of the idea is this: Suppose we ca prove that wheever oe of these statemets is true, the the oe followig it i the list is also true. I other words, For every k, if Pk is true, the Pk is true. This is called the iductio step because it leads us from the truth of oe statemet to the truth of the ext. Now suppose that we ca also prove that P is true. The iductio step ow leads us through the followig chai of statemets: P is true, so P is true. P is true, so P3 is true. P3 is true, so P4 is true... Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

34 86 CHAPTER Sequeces ad Series So we see that if both the iductio step ad P are proved, the statemet P is proved for all. Here is a summary of this importat method of proof. PRINCIPLE OF MATHEMATICAL INDUCTION For each atural umber, let P be a statemet depedig o. Suppose that the followig two coditios are satisfied.. P is true.. For every atural umber k, if Pk is true the Pk is true. The P is true for all atural umbers. To apply this priciple, there are two steps: Step Prove that P is true. Step Assume that Pk is true, ad use this assumptio to prove that Pk is true. Notice that i Step we do ot prove that Pk is true. We oly show that if Pk is true, the Pk is also true. The assumptio that Pk is true is called the iductio hypothesis. 979 Natioal Coucil of Teachers of Mathematics. Used by permissio. Courtesy of Adrejs Dukels, Swede. We ow use mathematical iductio to prove that the cojecture that we made at the begiig of this sectio is true. EXAMPLE A Proof by Mathematical Iductio Prove that for all atural umbers, SOLUTION Let P deote the statemet Step We eed to show that P is true. But P is simply the statemet that, which is of course true. Step We assume that Pk is true. Thus our iductio hypothesis is k k We wat to use this to show that Pk is true, that is, k 3k 4 k 3Note that we get Pk by substitutig k for each i the statemet P.4 We start with the left side ad use the iductio hypothesis to obtai the right side of the equatio: Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

35 SECTION.5 Mathematical Iductio 87 This equals k by the iductio hypothesis k 3k k 4 3k 4 Group the first k terms k 3k 4 Iductio hypothesis k 3k 4 Distributive Property k k Simplify k Factor Thus Pk follows from Pk, ad this completes the iductio step. Havig proved Steps ad, we coclude by the Priciple of Mathematical Iductio that P is true for all atural umbers. NOW TRY EXERCISE 3 EXAMPLE A Proof by Mathematical Iductio Prove that for every atural umber, 3... SOLUTION Let P be the statemet 3... /. We wat to show that P is true for all atural umbers. Step We eed to show that P is true. But P says that ad this statemet is clearly true. Step Assume that Pk is true. Thus our iductio hypothesis is 3... kk k We wat to use this to show that Pk is true, that is, 3... k 3k 4 k k So we start with the left side ad use the iductio hypothesis to obtai the right side: kk This equals by the iductio hypothesis 3... k k k4 k kk k k a k b Group the first k terms Iductio hypothesis Factor k k a k b k 3k 4 Commo deomiator Write k as k Thus Pk follows from Pk, ad this completes the iductio step. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

36 88 CHAPTER Sequeces ad Series Havig proved Steps ad, we coclude by the Priciple of Mathematical Iductio that P is true for all atural umbers. NOW TRY EXERCISE 5 The followig box gives formulas for the sums of powers of the first atural umbers. These formulas are importat i calculus. Formula is proved i Example. The other formulas are also proved by usig mathematical iductio (see Exercises 6 ad 9). SUMS OF POWERS 0. a. k. 3. a k 3 a k 4 k a k 6 k k It might happe that a statemet P is false for the first few atural umbers but true from some umber o. For example, we might wat to prove that P is true for 5. Notice that if we prove that P5 is true, the this fact, together with the iductio step, would imply the truth of P5, P6, P7,... The ext example illustrates this poit. EXAMPLE 3 Provig a Iequality by Mathematical Iductio Prove that 4 for all 5. SOLUTION Let P deote the statemet 4. Step P5 is the statemet that 4 # 5 5, or 0 3, which is true. Step Assume that Pk is true. Thus our iductio hypothesis is We get P(k ) by replacig by k i the statemet P(). 4k k We wat to use this to show that Pk is true, that is, 4k k The Art Archives/Corbis BLAISE PASCAL (63 66) is cosidered oe of the most versatile mids i moder history. He was a writer ad philosopher as well as a gifted mathematicia ad physicist. Amog his cotributios that appear i this book are Pascal s triagle ad the Priciple of Mathematical Iductio. Pascal s father, himself a mathematicia, believed that his so should ot study mathematics util he was 5 or 6. But at age, Blaise isisted o learig geometry ad proved most of its elemetary theorems himself. At 9 he iveted the first mechaical addig machie. I 647, after writig a major treatise o the coic sectios, he abruptly abadoed mathematics because he felt that his itese studies were cotributig to his ill health. He devoted himself istead to frivolous recreatios such as gamblig, but this oly served to pique his iterest i probability. I 654 he miraculously survived a carriage accidet i which his horses ra off a bridge. Takig this to be a sig from God, Pascal etered a moastery, where he pursued theology ad philosophy, writig his famous Pesées. He also cotiued his mathematical research. He valued faith ad ituitio more tha reaso as the source of truth, declarig that the heart has its ow reasos, which reaso caot kow. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

37 SECTION.5 Mathematical Iductio 89 So we start with the left-had side of the iequality ad use the iductio hypothesis to show that it is less tha the right-had side. For k 5 we have 4 k 4k 4 Distributive Property k 4 Iductio hypothesis k 4k Because 4 4k k k Iductio hypothesis # k k Property of expoets Thus Pk follows from Pk, ad this completes the iductio step. Havig proved Steps ad, we coclude by the Priciple of Mathematical Iductio that P is true for all atural umbers 5. NOW TRY EXERCISE.5 EXERCISES CONCEPTS. Mathematical iductio is a method of provig that a statemet P is true for all that is true. umbers. I Step we prove. Which of the followig is true about Step i a proof by mathematical iductio? (i) We prove Pk is true. (ii) We prove If Pk is true, the Pk is true. SKILLS 3 4 Use mathematical iductio to prove that the formula is true for all atural umbers # # 3 3 # # 3 # 4 3 # # # 3 3 # # # 3 # 3 4 # 4... # 5. Show that is divisible by for all atural umbers. 6. Show that 5 is divisible by 4 for all atural umbers. 7. Show that 4 is odd for all atural umbers. 8. Show that 3 3 is divisible by 3 for all atural umbers. 9. Show that 8 3 is divisible by 5 for all atural umbers. 0. Show that 3 is divisible by 8 for all atural umbers.. Prove that for all atural umbers.. Prove that for all atural umbers Prove that if x, the x x for all atural umbers. 4. Show that 00 for all Let a 3a ad a 5. Show that a 5 3 for all atural umbers. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

38 80 CHAPTER Sequeces ad Series 6. A sequece is defied recursively by a 3a 8 ad a 4. Fid a explicit formula for a, ad the use mathematical iductio to prove that the formula you foud is true. 7. Show that x y is a factor of x y for all atural umbers. 3Hit: x k y k x k x y x k y k y.4 8. Show that x y is a factor of x y for all atural umbers F deotes the th term of the Fiboacci sequece discussed i Sectio.. Use mathematical iductio to prove the statemet. 9. F 3 is eve for all atural umbers. 30. F F F 3... F F 3. F F F 3... F F F 3. F F 3... F F 33. For all, c 0 d c F 34. Let a be the th term of the sequece defied recursively by a a ad let a. Fid a formula for a i terms of the Fiboacci umbers F. Prove that the formula you foud is valid for all atural umbers. 35. Let F be the th term of the Fiboacci sequece. Fid ad prove a iequality relatig ad F for atural umbers. 36. Fid ad prove a iequality relatig 00 ad 3. F F F d thik it is false, give a example i which it fails. (a) p is prime for all. (b) for all. (c) is divisible by 3 for all. (d) 3 for all. (e) 3 is divisible by 3 for all. (f) 3 6 is divisible by 6 for all. 38. All Cats Are Black? What is wrog with the followig proof by mathematical iductio that all cats are black? Let P deote the statemet I ay group of cats, if oe cat is black, the they are all black. Step The statemet is clearly true for. Step Suppose that Pk is true. We show that Pk is true. Suppose we have a group of k cats, oe of whom is black; call this cat Tadpole. Remove some other cat (call it Sparky ) from the group. We are left with k cats, oe of whom (Tadpole) is black, so by the iductio hypothesis, all k of these are black. Now put Sparky back i the group ad take out Tadpole. We agai have a group of k cats, all of whom except possibly Sparky are black. The by the iductio hypothesis, Sparky must be black too. So all k cats i the origial group are black. Thus by iductio P is true for all. Sice everyoe has see at least oe black cat, it follows that all cats are black. DISCOVERY DISCUSSION WRITING 37. True or False? Determie whether each statemet is true or false. If you thik the statemet is true, prove it. If you Tadpole Sparky.6 THE BINOMIAL THEOREM Expadig (a b) The Biomial Coefficiets The Biomial Theorem Proof of the Biomial Theorem A expressio of the form a b is called a biomial. Although i priciple it s easy to raise a b to ay power, raisig it to a very high power would be tedious. I this sectio we fid a formula that gives the expasio of a b for ay atural umber ad the prove it usig mathematical iductio. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

39 SECTION.6 The Biomial Theorem 8 Expadig (a b) To fid a patter i the expasio of a b, we first look at some special cases. a b a b a b a ab b a b 3 a 3 3a b 3ab b 3 a b 4 a 4 4a 3 b 6a b 4ab 3 b 4 a b 5 a 5 5a 4 b 0a 3 b 0a b 3 5ab 4 b 5. The followig simple patters emerge for the expasio of a b.. There are terms, the first beig a ad the last beig b.. The expoets of a decrease by from term to term, while the expoets of b icrease by. 3. The sum of the expoets of a ad b i each term is. For istace, otice how the expoets of a ad b behave i the expasio of a b 5. The expoets of a decrease: The expoets of b icrease: With these observatios we ca write the form of the expasio of a b for ay atural umber. For example, writig a questio mark for the missig coefficiets, we have Óa bô 8 a 8? a 7 b? a 6 b? a 5 b 3? a 4 b 4? a 3 b 5? a b 6? ab 7 b 8 To complete the expasio, we eed to determie these coefficiets. To fid a patter, let s write the coefficiets i the expasio of a b for the first few values of i a triagular array as show i the followig array, which is called Pascal s triagle. a b 0 a b a b a b 3 a b 4 a b a b 5 a 5a b 0a b 0a b 3 5a b 4 b a b 5 a 5 5a 4 b 0a 3 b 0a b 5a b b The row correspodig to a b 0 is called the zeroth row ad is icluded to show the symmetry of the array. The key observatio about Pascal s triagle is the followig property KEY PROPERTY OF PASCAL S TRIANGLE Every etry (other tha a ) is the sum of the two etries diagoally above it. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

40 8 CHAPTER Sequeces ad Series What we ow call Pascal s triagle appears i this Chiese documet by Chu Shikie, dated 303. The title reads The Old Method Chart of the Seve Multiplyig Squares. The triagle was rediscovered by Pascal (see page 88). From this property it is easy to fid ay row of Pascal s triagle from the row above it. For istace, we fid the sixth ad seveth rows, startig with the fifth row: a b a b a b 7 To see why this property holds, let s cosider the followig expasios: a b 5 a 5 5a 4 b 0a 3 b 0a b 3 5ab 4 b 5 a b 6 a 6 6a 5 b 5a 4 b 0a 3 b 3 5a b 4 6ab 5 b 6 We arrive at the expasio of a b 6 by multiplyig a b 5 by a b. Notice, for istace, that the circled term i the expasio of a b 6 is obtaied via this multiplicatio from the two circled terms above it. We get this term whe the two terms above it are multiplied by b ad a, respectively. Thus its coefficiet is the sum of the coefficiets of these two terms. We will use this observatio at the ed of this sectio whe we prove the Biomial Theorem. Havig foud these patters, we ca ow easily obtai the expasio of ay biomial, at least to relatively small powers EXAMPLE Expadig a Biomial Usig Pascal s Triagle Fid the expasio of a b 7 usig Pascal s triagle. SOLUTION The first term i the expasio is a 7, ad the last term is b 7. Usig the fact that the expoet of a decreases by from term to term ad that of b icreases by from term to term, we have a b 7 a 7? a 6 b? a 5 b? a 4 b 3? a 3 b 4? a b 5? ab 6 b 7 The appropriate coefficiets appear i the seveth row of Pascal s triagle. Thus a b 7 a 7 7a 6 b a 5 b 35a 4 b 3 35a 3 b 4 a b 5 7ab 6 b 7 NOW TRY EXERCISE 5 EXAMPLE Expadig a Biomial Usig Pascal s Triagle Use Pascal s triagle to expad 3x 5. SOLUTION We fid the expasio of a b 5 ad the substitute for a ad 3x for b. Usig Pascal s triagle for the coefficiets, we get a b 5 a 5 5a 4 b 0a 3 b 0a b 3 5ab 4 b 5 Substitutig a ad b 3x gives 3x x 0 3 3x 0 3x 3 53x 4 3x x 70x 080x 3 80x 4 43x 5 NOW TRY EXERCISE 3 The Biomial Coefficiets Although Pascal s triagle is useful i fidig the biomial expasio for reasoably small values of, it is t practical for fidig a b for large values of. The reaso is that the method we use for fidig the successive rows of Pascal s triagle is recursive. Thus, to fid the 00th row of this triagle, we must first fid the precedig 99 rows. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

41 SECTION.6 The Biomial Theorem 83 4! # # 3 # 4 4 7! # # 3 # 4 # 5 # 6 # ! # # 3 # 4 # 5 # 6 # 7 # 8 # 9 # 0 3,68,800 We eed to examie the patter i the coefficiets more carefully to develop a formula that allows us to calculate directly ay coefficiet i the biomial expasio. Such a formula exists, ad the rest of this sectio is devoted to fidig ad provig it. However, to state this formula, we eed some otatio. The product of the first atural umbers is deoted by! ad is called factorial. We also defie 0! as follows:! # # 3 #... # # 0! This defiitio of 0! makes may formulas ivolvig factorials shorter ad easier to write. THE BINOMIAL COEFFICIENT Let ad r be oegative itegers with r. The biomial coefficiet is deoted by ()ad is defied by r a r b! r! r! EXAMPLE 3 (a) (b) (c) Calculatig Biomial Coefficiets a 9 4 b 9! 9! 4!9 4! 4! 5! # # 3 # 4 # 5 # 6 # 7 # 8 # 9 # # 3 # 4 # # 3 # 4 # 5 a 00 3 b 00! 3!00 3! # # 3 # p # 97 # 98 # 99 # 00 # # 3 # # 3 # p # 97 a b 00! 97!00 97! # # 3 # p # 97 # 98 # 99 # 00 # # 3 # p # 97 # # 3 NOW TRY EXERCISES 7 AND 9 6 # 7 # 8 # 9 # # 3 # # 99 # 00 # # 3 6, # 99 # 00 # # 3 6,700 Although the biomial coefficiet ( r )is defied i terms of a fractio, all the results of Example 3 are atural umbers. I fact, ()is r always a atural umber (see Exercise 54). Notice that the biomial coefficiets i parts (b) ad (c) of Example 3 are equal. This is a special case of the followig relatio, which you are asked to prove i Exercise 5. a r b a r b Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

42 84 CHAPTER Sequeces ad Series To see the coectio betwee the biomial coefficiets ad the biomial expasio of a b, let s calculate the followig biomial coefficiets: a 5 b 5! 0!5! a 5 0 b a 5 b 5 a 5 b 0 a 5 3 b 0 a 5 4 b 5 a 5 5 b These are precisely the etries i the fifth row of Pascal s triagle. I fact, we ca write Pascal s triagle as follows. a 0 0 b a 0 b a b a 0 b a b a b a 3 0 b a 3 b a 3 b a 3 3 b a 4 0 b a 4 b a 4 b a 4 3 b a 4 4 b a 5 0 b a 5 b a 5 b a 5 3 b a 5 4 b a 5 5 b # # # # # # # a 0 b a b a b # # # a b To demostrate that this patter holds, we eed to show that ay etry i this versio of Pascal s triagle is the sum of the two etries diagoally above it. I other words, we must show that each etry satisfies the key property of Pascal s triagle. We ow state this property i terms of the biomial coefficiets. a b KEY PROPERTY OF THE BINOMIAL COEFFICIENTS For ay oegative itegers r ad k with r k, k a r b a k k b a b r r Notice that the two terms o the left-had side of this equatio are adjacet etries i the kth row of Pascal s triagle ad the term o the right-had side is the etry diagoally below them, i the k st row. Thus this equatio is a restatemet of the key property of Pascal s triagle i terms of the biomial coefficiets. A proof of this formula is outlied i Exercise 53. The Biomial Theorem We are ow ready to state the Biomial Theorem. THE BINOMIAL THEOREM a b a 0 b a a b a b a b a b... a b ab a b b Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

43 SECTION.6 The Biomial Theorem 85 We prove this theorem at the ed of this sectio. First, let s look at some of its applicatios. EXAMPLE 4 Expadig a Biomial Usig the Biomial Theorem Use the Biomial Theorem to expad x y 4. SOLUTION Verify that By the Biomial Theorem, x y 4 a 4 0 b x 4 a 4 b x 3 y a 4 b x y a 4 3 b xy 3 a 4 4 b y 4 a 4 0 b a 4 b 4 a 4 b 6 a 4 3 b 4 a 4 4 b It follows that NOW TRY EXERCISE 5 x y 4 x 4 4x 3 y 6x y 4xy 3 y 4 EXAMPLE 5 Expadig a Biomial Usig the Biomial Theorem Use the Biomial Theorem to expad A x B 8. SOLUTION We first fid the expasio of a b 8 ad the substitute x for a ad for b. Usig the Biomial Theorem, we have a b 8 a 8 0 b a8 a 8 b a7 b a 8 b a6 b a 8 3 b a 5 b 3 a 8 4 b a 4 b 4 Verify that a 8 5 b a 3 b 5 a 8 6 b a b 6 a 8 7 b ab7 a 8 8 b b8 a 8 0 b a 8 b 8 a 8 b 8 a 8 3 b 56 a 8 b 70 4 a 8 5 b 56 a 8 6 b 8 a 8 7 b 8 a 8 8 b So a b 8 a 8 8a 7 b 8a 6 b 56a 5 b 3 70a 4 b 4 56a 3 b 5 8a b 6 8ab 7 b 8 Performig the substitutios a x / ad b gives A x B 8 x / 8 8x / 7 8x / 6 56x / x / x / 3 5 8x / 6 8x / 7 8 This simplifies to x 8 x 4 8x 7/ 8x 3 56x 5/ 70x 56x 3/ 8x 8x / NOW TRY EXERCISE 7 The Biomial Theorem ca be used to fid a particular term of a biomial expasio without havig to fid the etire expasio. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

44 86 CHAPTER Sequeces ad Series GENERAL TERM OF THE BINOMIAL EXPANSION The term that cotais a r i the expasio of a b is a r b ar b r EXAMPLE 6 Fidig a Particular Term i a Biomial Expasio Fid the term that cotais x 5 i the expasio of x y 0. SOLUTION The term that cotais x 5 is give by the formula for the geeral term with a x, b y, 0, ad r 5. So this term is a 0 5 b a 5 b 5 0! 5!0 5! x5 y 5 0! 5! 5! 3x 5 y 5 496,8x 5 y 5 NOW TRY EXERCISE 39 EXAMPLE 7 Fidig a Particular Term i a Biomial Expasio Fid the coefficiet of x 8 i the expasio of a x 0. x b SOLUTION Both x ad /x are powers of x, so the power of x i each term of the expasio is determied by both terms of the biomial. To fid the required coefficiet, we first fid the geeral term i the expasio. By the formula we have a x, b /x, ad 0, so the geeral term is 0 a 0 r bx r a 0r x b 0 a 0 r b x r x 0r 0 a 0 r b x 3r0 Thus the term that cotais x 8 is the term i which 3r 0 8 r 6 So the required coefficiet is 0 a 0 6 b a 0 b 0 4 NOW TRY EXERCISE 4 Proof of the Biomial Theorem We ow give a proof of the Biomial Theorem usig mathematical iductio. PROOF Let P deote the statemet a b a 0 b a a b a b a b a b... a b ab a b b Step We show that P is true. But P is just the statemet a b a 0 b a a b b a b a b which is certaily true. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

45 SECTION.6 The Biomial Theorem 87 Step We assume that Pk is true. Thus our iductio hypothesis is a b k a k 0 b a k a k b a k b a k b a k b... k a k b ab k a k k b b k We use this to show that Pk is true. a b k a b3a b k 4 a b ca k 0 b a k a k b a k b a k b a k b... k a k b ab k a k k b b k d a ca k 0 b a k a k b a k b a k b a k b... k a k b ab k a k k b b k d b ca k 0 b a k a k b a k b a k b a k b... k a k b ab k a k k b b k d a k 0 b a k a k b a k b a k b a k b... k a k b a b k a k k b ab k a k 0 b a k b a k b a k b a k b a k b 3... k a k b ab k a k k b b k Iductio hypothesis Distributive Property Distributive Property a k 0 b a k ca k 0 b a k bda k b ca k b a k bda k b a b k a k 0 Group... k ca k b a k k bdab k a k k b b k like terms Usig the key property of the biomial coefficiets, we ca write each of the expressios i square brackets as a sigle biomial coefficiet. Also, writig k k the first ad last coefficiets as ( ) ad ( ) (these are equal to by Exercise 50) gives b a k a k b a k b a k 0 k b a k b... a k b ab k a k k k b b k But this last equatio is precisely Pk, ad this completes the iductio step. Havig proved Steps ad, we coclude by the Priciple of Mathematical Iductio that the theorem is true for all atural umbers..6 EXERCISES CONCEPTS. A algebraic expressio of the form a b, which cosists of a sum of two terms, is called a.. We ca fid the coefficiets i the expasio of a b from the th row of triagle. So a b 4 a 4 a 3 b a b ab 3 b 4 3. The biomial coefficiets ca be calculated directly by usig the formula a. So a 4. k b 3 b 4. To expad a b, we ca use the Theorem. Usig this theorem, we fid a b 4 a b a 4 a b a 3 b a b a b a b ab 3 a b b 4 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

46 88 CHAPTER Sequeces ad Series SKILLS 5 6 Use Pascal s triagle to expad the expressio a x 4 x y 6 x 4 x b 8. x y 5 9. x 5 0. A a bb 6. x. A B 6 3. x 3y a x 5 a 5 x 3 3 x x b b 7 4 Evaluate the expressio. 7. a 6 8. a b 3 b 0. a 0. a 3. a 5 5 b ba4 b ba5 3 b a 5 0 b a 5 b a 5 b a 5 3 b a 5 4 b a 5 5 b a 5 0 b a 5 b a 5 b a 5 3 b a 5 4 b a 5 5 b 5 8 Use the Biomial Theorem to expad the expressio. 5. x y 4 6. x 5 7. a 6 8. A B 4 x b 9. Fid the first three terms i the expasio of x y Fid the first four terms i the expasio of x / Fid the last two terms i the expasio of a /3 a / Fid the first three terms i the expasio of a x x b Fid the middle term i the expasio of x Fid the fifth term i the expasio of ab Fid the 4th term i the expasio of a b Fid the 8th term i the expasio of A B Fid the 00th term i the expasio of y Fid the secod term i the expasio of a x x b Fid the term cotaiig x 4 i the expasio of x y Fid the term cotaiig y 3 i the expasio of A yb. 4. Fid the term cotaiig b 8 i the expasio of a b. 4. Fid the term that does ot cotai x i the expasio of a 8x x b Factor usig the Biomial Theorem. 43. x 4 4x 3 y 6x y 4xy 3 y 4 a b 44. x 5 5x 4 0x 3 0x 5x 45. 8a 3 a b 6ab b x 8 4x 6 y 6x 4 y 4x y 3 y Simplify usig the Biomial Theorem. x h 3 x 3 x h 4 x h h 49. Show that [Hit: Note that , ad use the Biomial Theorem to show that the sum of the first two terms of the expasio is greater tha.] 50. Show that a ad a. 0 b b 5. Show that a. b a b 5. Show that a for 0 r. r b a r b 53. I this exercise we prove the idetity a r b a b a b r r (a) Write the left-had side of this equatio as the sum of two fractios. (b) Show that a commo deomiator of the expressio that you foud i part (a) is r! r!. (c) Add the two fractios usig the commo deomiator i part (b), simplify the umerator, ad ote that the resultig expressio is equal to the right-had side of the equatio. 54. Prove that r is a iteger for all ad for 0 r. [Suggestio: Use iductio to show that the statemet is true for all, ad use Exercise 53 for the iductio step.] APPLICATIONS 55. Differece i Volumes of Cubes The volume of a cube of side x iches is give by Vx x 3, so the volume of a cube of side x iches is give by Vx x 3. Use the Biomial Theorem to show that the differece i volume betwee the larger ad smaller cubes is 6x x 8 cubic iches. 56. Probability of Hittig a Target The probability that a archer hits the target is p 0.9, so the probability that he misses the target is q 0.. It is kow that i this situatio the probability that the archer hits the target exactly r times i attempts is give by the term cotaiig p r i the biomial expasio of p q. Fid the probability that the archer hits the target exactly three times i five attempts. DISCOVERY DISCUSSION WRITING 57. Powers of Factorials Which is larger, 00! 0 or 0! 00? [Hit: Try factorig the expressios. Do they have ay commo factors?] Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

47 CHAPTER Review Sums of Biomial Coefficiets Add each of the first five rows of Pascal s triagle, as idicated. Do you see a patter??? 3 3? 4 6 4? ? O the basis of the patter you have foud, fid the sum of the th row: 59. Alteratig Sums of Biomial Coefficiets Fid the sum a 0 b a b a b... a b by fidig a patter as i Exercise 58. Prove your result by expadig usig the Biomial Theorem. a 0 b a b a b... a b Prove your result by expadig usig the Biomial Theorem. CHAPTER REVIEW CONCEPT CHECK. (a) What is a sequece? (b) What is a arithmetic sequece? Write a expressio for the th term of a arithmetic sequece. (c) What is a geometric sequece? Write a expressio for the th term of a geometric sequece.. (a) What is a recursively defied sequece? (b) What is the Fiboacci sequece? 3. (a) What is meat by the partial sums of a sequece? (b) If a arithmetic sequece has first term a ad commo differece d, write a expressio for the sum of its first terms. (c) If a geometric sequece has first term a ad commo ratio r, write a expressio for the sum of its first terms. (d) Write a expressio for the sum of a ifiite geometric series with first term a ad commo ratio r. For what values of r is your formula valid? 4. (a) Write the sum a a k without usig sigma otatio. k (b) Write b b b 3... b usig sigma otatio. 5. Write a expressio for the amout A f of a auity cosistig of regular equal paymets of size R with iterest rate i per time period. 6. State the Priciple of Mathematical Iductio. 7. Write the first five rows of Pascal s triagle. How are the etries related to each other? 8. (a) What does the symbol! mea? (b) Write a expressio for the biomial coefficiet r. (c) State the Biomial Theorem. (d) Write the term that cotais a r i the expasio of a b. EXERCISES 6 Fid the first four terms as well as the teth term of the sequece with the give th term.. a. a 3. a 4. a 3 5. a 6. a a! b! 7 0 A sequece is defied recursively. Fid the first seve terms of the sequece. 7. a a, a 8. a a, a 9. a a a, a, a 3 0. a 3a, a 3 Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

48 830 CHAPTER Sequeces ad Series 4 The th term of a sequece is give. (a) Fid the first five terms of the sequece. (b) Graph the terms you foud i part (a). (c) Fid the fifth partial sum of the sequece. (d) Determie whether the series is arithmetic or geometric. Fid the commo differece or the commo ratio.. a 5. a a 4. a 4 5 The first four terms of a sequece are give. Determie whether they ca be the terms of a arithmetic sequece, a geometric sequece, or either. If the sequece is arithmetic or geometric, fid the fifth term. 5. 5, 5.5, 6, 6.5, t 3, t, t, t, a,, a, 4,, 3, 9,... a, Show that 3, 6i,, 4i,... is a geometric sequece, ad fid the commo ratio. (Here i.) 4. Fid the th term of the geometric sequece, i, 4i, 4 4i, 8,... (Here i.) 5. The sixth term of a arithmetic sequece is 7, ad the fourth term is. Fid the secod term. 6. The 0th term of a arithmetic sequece is 96, ad the commo differece is 5. Fid the th term. 7. The third term of a geometric sequece is 9, ad the commo 3 ratio is. Fid the fifth term. 8. The secod term of a geometric sequece is 0, ad the fifth 50 term is. Fid the th term. 7,, 3, 4,...,,, 4, t 3, t 3 5, t,,... 0.,,,, A teacher makes $3,000 i his first year at Lakeside School ad gets a 5% raise each year. (a) Fid a formula for his salary A i his th year at this school. (b) List his salaries for his first 8 years at this school. 30. A colleague of the teacher i Exercise 9, hired at the same time, makes $35,000 i her first year, ad gets a $00 raise each year. (a) What is her salary A i her th year at this school? (b) Fid her salary i her eighth year at this school, ad compare it to the salary of the teacher i Exercise 9 i his eighth year. 3. A certai type of bacteria divides every 5 s. If three of these bacteria are put ito a petri dish, how may bacteria are i the dish at the ed of mi? 3. If a, a, a 3,... ad b, b, b 3,... are arithmetic sequeces, show that a b, a b, a 3 b 3,... is also a arithmetic sequece. 33. If a, a, a 3,... ad b, b, b 3,... are geometric sequeces, show that a b, a b, a 3 b 3,... is also a geometric sequece. 34. (a) If a, a, a 3,... is a arithmetic sequece, is the sequece a, a, a 3,... arithmetic? (b) If a, a, a 3,... is a geometric sequece, is the sequece 5a,5a,5a 3,... geometric? 35. Fid the values of x for which the sequece 6, x,,... is (a) arithmetic (b) geometric 36. Fid the values of x ad y for which the sequece, x, y, 7,... is (a) arithmetic (b) geometric Fid the sum. 37. a k 38. a 39. a k k 40. a 4 44 Write the sum without usig sigma otatio. Do ot evaluate a k 4. a j 43. a 44. a Write the sum usig sigma otatio. Do ot evaluate k3 6 k k 50 3 k k k # # 3 3 # Determie whether the expressio is a partial sum of a arithmetic or geometric sequece. The fid the sum # 53. a a 75 k/ 0 # 999 # Determie whether the ifiite geometric series is coverget or diverget. If it is coverget, fid its sum / / a 9 8 b a b a ab ab 4 ab 6..., 0 b 0 # # 4 i 5 3 m m j 0 8 k0 i i # Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

49 CHAPTER Review The first term of a arithmetic sequece is a 7, ad the commo differece is d 3. How may terms of this sequece must be added to obtai 35? 6. The sum of the first three terms of a geometric series is 5, ad the commo ratio is r 3. Fid the first term. 63. A perso has two parets, four gradparets, eight greatgradparets, ad so o. What is the total umber of a perso s acestors i 5 geeratios? 64. Fid the amout of a auity cosistig of 6 aual paymets of $000 each ito a accout that pays 8% iterest per year, compouded aually. 65. How much moey should be ivested every quarter at % per year, compouded quarterly, i order to have $0,000 i oe year? 66. What are the mothly paymets o a mortgage of $60,000 at 9% iterest if the loa is to be repaid i (a) 30 years? (b) 5 years? Use mathematical iductio to prove that the formula is true for all atural umbers # 3 3 # 5 5 # a ba ba 3 b... a b 70. Show that 7 is divisible by 6 for all atural umbers. 7. Let a 3a 4 ad a 4. Show that a # 3 for all atural umbers. 7. Prove that the Fiboacci umber F 4 is divisible by 3 for all atural umbers Evaluate the expressio. 73. a a 0 b a 0 ba5 3 b 6 b a a k b k Expad the expressio. 77. A B x x x y 4 8. Fid the 0th term i the expasio of a b. 8. Fid the first three terms i the expasio of b /3 b / Fid the term cotaiig A 6 i the expasio of A 3B 0. 8 a k0 a 8 k ba 8 8 k b Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

50 CHAPTER TEST. Fid the first six terms ad the sixth partial sum of the sequece whose th term is a.. A sequece is defied recursively by a 3a, a. Fid the first six terms of the sequece. 3. A arithmetic sequece begis, 5, 8,, 4,... (a) Fid the commo differece d for this sequece. (b) Fid a formula for the th term a of the sequece. (c) Fid the 35th term of the sequece. 4. A geometric sequece begis, 3, 3/4, 3/6, 3/64,... (a) Fid the commo ratio r for this sequece. (b) Fid a formula for the th term a of the sequece. (c) Fid the teth term of the sequece. 5. The first term of a geometric sequece is 5, ad the fourth term is 5. (a) Fid the commo ratio r ad the fifth term. (b) Fid the partial sum of the first eight terms. 6. The first term of a arithmetic sequece is 0, ad the teth term is. (a) Fid the commo differece ad the 00th term of the sequece. (b) Fid the partial sum of the first te terms. 7. Let a, a, a 3,... be a geometric sequece with iitial term a ad commo ratio r. Show that a, a, a 3,... is also a geometric sequece by fidig its commo ratio. 8. Write the expressio without usig sigma otatio, ad the fid the sum. (a) 5 a 9. Fid the sum. (a) (b) /... 3/ 0. Use mathematical iductio to prove that for all atural umbers, (b) a. Expad x y 5.. Fid the term cotaiig x 3 i the biomial expasio of 3x A puppy weighs 0.85 lb at birth, ad each week he gais 4% i weight. Let a be his weight i pouds at the ed of his th week of life. (a) Fid a formula for a. (b) How much does the puppy weigh whe he is six weeks old? (c) Is the sequece a, a, a 3,... arithmetic, geometric, or either? Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

51 FOCUS ON MODELING Modelig with Recursive Sequeces May real-world processes occur i stages. Populatio growth ca be viewed i stages each ew geeratio represets a ew stage i populatio growth. Compoud iterest is paid i stages each iterest paymet creates a ew accout balace. May thigs that chage cotiuously are more easily measured i discrete stages. For example, we ca measure the temperature of a cotiuously coolig object i oe-hour itervals. I this Focus we lear how recursive sequeces are used to model such situatios. I some cases we ca get a explicit formula for a sequece from the recursio relatio that defies it by fidig a patter i the terms of the sequece. Recursive Sequeces as Models Suppose you deposit some moey i a accout that pays 6% iterest compouded mothly. The bak has a defiite rule for payig iterest: At the ed of each moth the bak adds to your accout % (or 0.005) of the amout i your accout at that time. Let s express this rule as follows: amout at the ed of this moth amout at the ed of last moth amout at the ed of last moth Usig the Distributive Property, we ca write this as amout at the ed of this moth.005 amout at the ed of last moth To model this statemet usig algebra, let A 0 be the amout of the origial deposit, let A be the amout at the ed of the first moth, let A be the amout at the ed of the secod moth, ad so o. So A is the amout at the ed of the th moth. Thus A.005A We recogize this as a recursively defied sequece it gives us the amout at each stage i terms of the amout at the precedig stage A A 0 A A A To fid a formula for A, let s fid the first few terms of the sequece ad look for a patter. A.005A 0 We see that i geeral, A.005 A 0. A.005A.005 A 0 A 3.005A A 0 A 4.005A A Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

52 834 Focus o Modelig EXAMPLE Populatio Growth A certai aimal populatio grows by % each year. The iitial populatio is (a) Fid a recursive sequece that models the populatio P at the ed of the th year. (b) Fid the first five terms of the sequece P. (c) Fid a formula for P. SOLUTION (a) We ca model the populatio usig the followig rule: populatio at the ed of this year.0 populatio at the ed of last year Algebraically, we ca write this as the recursio relatio P.0P (b) Sice the iitial populatio is 5000, we have P P.0P P.0P P 3.0P P 4.0P (c) We see from the patter exhibited i part (b) that P (Note that P is a geometric sequece, with commo ratio r.0.) EXAMPLE Daily Drug Dose A patiet is to take a 50-mg pill of a certai drug every morig. It is kow that the body elimiates 40% of the drug every 4 hours. (a) Fid a recursive sequece that models the amout A of the drug i the patiet s body after each pill is take. (b) Fid the first four terms of the sequece A. (c) Fid a formula for A. (d) How much of the drug remais i the patiet s body after 5 days? How much will accumulate i his system after prologed use? SOLUTION (a) Each morig, 60% of the drug remais i his system, plus he takes a additioal 50 mg (his daily dose). amout of drug this morig 0.6 amout of drug yesterday morig 50 mg Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

53 Modelig with Recursive Sequeces 835 We ca express this as a recursio relatio A 0.6A 50 (b) Sice the iitial dose is 50 mg, we have A 0 50 A 0.6A A 0.6A A 3 0.6A (c) From the patter i part (b) we see that A a 0.6 b Partial sum of a geometric sequece (page 80) Simplify (d) To fid the amout remaiig after 5 days, we substitute 5 ad get A mg. To fid the amout remaiig after prologed use, we let become large. As gets large, 0.6 approaches 0. That is, as q (see Sectio 4.). So as q, A Thus after prologed use the amout of drug i the patiet s system approaches 5 mg (see Figure, where we have used a graphig calculator to graph the sequece). Plot Plot Plot3 Mi=0 u( )=5(-.6^( +)) 50 Eter sequece 0 Graph sequece 6 FIGURE PROBLEMS. Retiremet Accouts May college professors keep retiremet savigs with TIAA, the largest auity program i the world. Iterest o these accouts is compouded ad credited daily. Professor Brow has $75,000 o deposit with TIAA at the start of 0 ad receives 3.65% iterest per year o his accout. (a) Fid a recursive sequece that models the amout A i his accout at the ed of the th day of 0. (b) Fid the first eight terms of the sequece A, rouded to the earest cet. (c) Fid a formula for A. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).

54 836 Focus o Modelig. Fitess Program Sheila decides to embark o a swimmig program as the best way to maitai cardiovascular health. She begis by swimmig 5 mi o the first day, the adds mi every day after that. (a) Fid a recursive formula for the umber of miutes T that she swims o the th day of her program. (b) Fid the first 6 terms of the sequece T. (c) Fid a formula for T. What kid of sequece is this? (d) O what day does Sheila attai her goal of swimmig at least 65 mi a day? (e) What is the total amout of time she will have swum after 30 days? 3. Mothly Savigs Program Alice opes a savigs accout that pays 3% iterest per year, compouded mothly. She begis by depositig $00 at the start of the first moth ad adds $00 at the ed of each moth, whe the iterest is credited. (a) Fid a recursive formula for the amout A i her accout at the ed of the th moth. (Iclude the iterest credited for that moth ad her mothly deposit.) (b) Fid the first five terms of the sequece A. (c) Use the patter you observed i (b) to fid a formula for A. [Hit: To fid the patter most easily, it s best ot to simplify the terms too much.] (d) How much has she saved after 5 years? 4. Stockig a Fish Pod A pod is stocked with 4000 trout, ad through reproductio the populatio icreases by 0% per year. Fid a recursive sequece that models the trout populatio P at the ed of the th year uder each of the followig circumstaces. Fid the trout populatio at the ed of the fifth year i each case. (a) The trout populatio chages oly because of reproductio. (b) Each year 600 trout are harvested. (c) Each year 50 additioal trout are itroduced ito the pod. (d) Each year 0% of the trout are harvested, ad 300 additioal trout are itroduced ito the pod. 5. Pollutio A chemical plat discharges 400 tos of pollutats every year ito a adjacet lake. Through atural ruoff, 70% of the pollutats cotaied i the lake at the begiig of the year are expelled by the ed of the year. (a) Explai why the followig sequece models the amout A of the pollutat i the lake at the ed of the th year that the plat is operatig. A 0.30A 400 (b) Fid the first five terms of the sequece A. Copyright 0 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s).