12.1 Sequences. Write the First Several Terms of a Sequence. PREPARING FOR THIS SECTION Before g etting started, review the following concept: ( )

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1 SECTION. Sequeces 79. Sequeces PREPARING FOR THIS SECTION Before g ettig started, review the followig cocept: Fuctios (Sectio., pp. 5) Now Work the Are You Prepared? problems o page 799. OBJECTIVES Write the First Several Terms of a Sequece (p. 79) Write the Terms of a Sequece Defied by a Recursive Formula (p. 79) Use Summatio Notatio (p. 797) Fid the Sum of a Sequece (p. 798) Whe you hear the word sequece as i a sequece of evets, you likely thik of somethig that happes first, the secod, ad so o. I mathematics, the word sequece also deals with outcomes that are first, secod, ad so o. DEFINITION A sequece is a fuctio whose domai is the set of positive itegers. So i a sequece the iputs are,,,...because a sequece is a fuctio, it will have a graph. I Figure (a), we show the graph of the fuctio fx = x, x 7 0. If all the poits o this graph were removed except those whose x -coordiates are positive itegers, that is, if all poits were removed except,, a, b, a, b, ad so o, the remaiig poits would be the graph of the sequece f = as, show i Figure (b). Notice that we use to represet the idepedet variable i a sequece. This serves to remid us that is a positive iteger. Figure y (, ), ( ) (, ) (, ) f( ) (, ) (, ) (, ), ( ) x (a) f(x) x, x 0 (b) f(), a positive iteger Write the First Several Terms of a Sequece A sequece is usually represeted by listig its values i order. For example, the sequece whose graph is give i Figure (b) might be represeted as f, f, f, f, Á or,,,, Á The list ever eds, as the ellipsis idicates. The umbers i this ordered list are called the terms of the sequece. I dealig with sequeces, we usually use subscripted letters, such as a, to represet the first term, a for the secod term, a for the third term, ad so o. For the sequece f = we write, a f(), first term a f(), secod term a f(), third term a f(), fourth term a f(), th term

2 79 CHAPTER Sequeces; Iductio; the Biomial Theorem I other words, we usually do ot use the traditioal fuctio otatio f for sequeces. For this particular sequece, we have a rule for the th term, which is a = so it is easy to fid ay term of the sequece., Whe a formula for the th term (sometimes called the geeral term) of a sequece is kow, rather tha write out the terms of the sequece, we usually represet the etire sequece by placig braces aroud the formula for the th term. For example, the sequece whose th term is b = a may be represeted as b or by 5b = ea b f b =, b =, b = 8, Á, b = a b, Á Figure a Solutio ( ) ( ) ( ) EXAMPLE, 5, 5,, 5 (, ) ( ) (, 0) 5 Writig the First Several Terms of a Sequece Write dow the first six terms of the followig sequece ad graph it. The first six terms of the sequece are See Figure for the graph. 5a = e - f a = 0, a =, a =, a =, a 5 = 5, a = 5 COMMENT Graphig utilities ca be used to write the terms of a sequece ad graph them. Figure shows the sequece give i Example geerated o a TI-8 Plus graphig calculator. We ca see the first few terms of the sequece o the viewig widow. You eed to press the right arrow key to scroll right to see the remaiig terms of the sequece. Figure shows a graph of the sequece. Notice that the first term of the sequece is ot visible sice it lies o the x-axis. TRACEig the graph will allow you to see the terms of the sequece. The TABLE feature ca also be used to geerate the terms of the sequece. See Table. Figure Figure Table Now Work PROBLEM 7 EXAMPLE Writig the First Several Terms of a Sequece Write dow the first six terms of the followig sequece ad graph it. 5b = e- + a bf

3 SECTION. Sequeces 795 Figure 5 Solutio The first six terms of the sequece are b (, ) b =, b = -, b =, b =-, b 5 = 5, b =- ( ), ( ) 5, 5 See Figure 5 for the graph. 5,, (, ) ( ) ( ) Notice i the sequece 5b i Example that the sigs of the terms alterate. This occurs whe we use factors such as - +, which equals if is odd ad - if is eve, or -, which equals - if is odd ad if is eve. EXAMPLE Writig the First Several Terms of a Sequece Write dow the first six terms of the followig sequece ad graph it. Solutio 5c = c The first six terms of the sequece are if is eve s if is odd Figure c 5 (, ) (, ) (, ) (, ) (, ) ( 5, 5 ) 5 See Figure for the graph. c =, c =, c =, c =, c 5 = 5, c = Now Work PROBLEM 9 Sometimes a sequece is idicated by a observed patter i the first few terms that makes it possible to ifer the makeup of the th term. I the example that follows, a sufficiet umber of terms of the sequece is give so that a atural choice for the th term is suggested. EXAMPLE Determiig a Sequece from a Patter (a) e, (b),, 9, 7, Á (c),, 5, 7, Á (d) (e), e, -, e,,, 9,, 5, Á, e, Á -, 5, Á Now Work PROBLEM 7 The Factorial Symbol a = e b = c = - d = - e = - - a b Some sequeces i mathematics ivolve a special product called a factorial. DEFINITION If Ú 0 is a iteger, the factorial symbol! is defied as follows: 0! =! =! = - # Á # # # if Ú

4 79 CHAPTER Sequeces; Iductio; the Biomial Theorem Table! For example,! = # =, Table lists the values of! for 0. Because we ca use the formula! = # # =,! = # # # =,! ( )( ) ( )!! = -! ad so o. to fid successive factorials. For example, because! = 70, we have Exploratio 7! = 7 #! = 770 = 500 Your calculator has a factorial key. Use it to see how fast factorials icrease i ad value. Fid the value of 9!. What 8! = 8 # 7! = 8500 = 0,0 happes whe you try to fid 70!? I fact, 70! is larger tha 0 00 (a googol). Now Work PROBLEM Write the Terms of a Sequece Defied by a Recursive Formula A secod way of defiig a sequece is to assig a value to the first (or the first few) term(s) ad specify the th term by a formula or equatio that ivolves oe or more of the terms precedig it. Sequeces defied this way are said to be defied recursively, ad the rule or formula is called a recursive formula. EXAMPLE 5 Solutio Writig the Terms of a Recursively Defied Sequece Write dow the first five terms of the followig recursively defied sequece. s =, s = s - The first term is give as s =. To get the secod term, use = i the formula s = s - to get s = s = # =. To get the third term, use = i the formula to get s = s = # =. To get a ew term requires that we kow the value of the precedig term. The first five terms are s = s = # = s = # = s = # = s 5 = 5 # = 0 Do you recogize this sequece? s =! EXAMPLE Solutio Writig the Terms of a Recursively Defied Sequece Write dow the first five terms of the followig recursively defied sequece. u =, u =, u = u - + u - We are give the first two terms.to get the third term requires that we kow both of the previous two terms. That is, u = u = u = u + u = + = u = u + u = + = u 5 = u + u = + = 5

5 SECTION. Sequeces 797 The sequece defied i Example is called the Fiboacci sequece, ad the terms of this sequece are called Fiboacci umbers. These umbers appear i a wide variety of applicatios (see Problems 85 88). Now Work PROBLEMS 5 AND Use Summatio Notatio It is ofte importat to fid the sum of the first terms of a sequece 5a, that is, a + a + a + Á + a Rather tha write dow all these terms, we itroduce a more cocise way to express the sum, called summatio otatio. Usig summatio otatio, we write the sum as a + a + a + Á + a = a The symbol (the Greek letter sigma, which is a S i our alphabet) is simply a istructio to sum, or add up, the terms. The iteger k is called the idex of the sum; it tells you where to start the sum ad where to ed it. The expressio a a k a k is a istructio to add the terms a k of the sequece 5a startig with ad edig with k =. We read the expressio as the sum of from to k =. a k EXAMPLE 7 Expadig Summatio Notatio Write out each sum. (a) a k (b) a k! Solutio (a) a Á + (b) a k! =! +! + Á +! Now Work PROBLEM 5 EXAMPLE 8 Solutio Writig a Sum i Summatio Notatio Express each sum usig summatio otatio. (a) Á + 9 (b) + (a) The sum Á + 9 has 9 terms, each of the form k, ad starts at ad eds at k = 9: (b) The sum Á + - has terms, each of the form ad starts at ad eds at k = : k -, Á + 9 = a Á Á + - k - = a k -

6 798 CHAPTER Sequeces; Iductio; the Biomial Theorem The idex of summatio eed ot always begi at or ed at ; for example, we could have expressed the sum i Example 8(b) as - Letters other tha k may be used as the idex. For example, each represet the same sum as the oe give i Example 7(b). j = Now Work PROBLEM a Á + a j! ad a i! i = - Fid the Sum of a Sequece Next we list some properties of sequeces usig summatio otatio. These properties are useful for addig the terms of a sequece. THEOREM Properties of Sequeces If 5a ad 5b are two sequeces ad c is a real umber, the: a ca k = ca + ca + Á + ca = ca + a + Á + a = c a a a k + a a k - a k = j + b k = a a k + b k = a a k - a k = a a k - j a a k where 0 j a b k a b k a k () () () () The proof of property () follows from the distributive property of real umbers. The proofs of properties ad are based o the commutative ad associative properties of real umbers. Property () states that the sum from j + to equals the sum from to mius the sum from to j. It ca be helpful to employ this property whe the idex of summatio begis at a umber larger tha. THEOREM Formulas for Sums of Sequeces a c c c... c c k a + + a k = Á + = + a k = Á + = c terms Á + = + c is a real umber d (5) () (7) (8) The proof of formula (5) follows from the defiitio of summatio otatio. You are asked to prove formula () i Problem 9. The proofs of formulas (7) ad (8) require mathematical iductio, which is discussed i Sectio..

7 SECTION. Sequeces 799 Notice the differece betwee formulas (5) ad (). I (5), the costat c is beig summed from to, while i () the idex of summatio k is beig summed from to. EXAMPLE 9 Solutio Fidig the Sum of a Sequece Fid the sum of each sequece. (a) 5 a k 5 5 (a) a k = a k 55 + = a b Property () Formula () (b) a k + = a k + a Property () 00 + = a b + 0 Formulas (8) ad (5) (c) a k - 7k + = a k - a 7k + a Properties () ad () = a k - = + # + = = 88 = 5 = 5 = = 05 7 a k + a (b) 0 (c) a k - 7k + (d) a k 0 a k + k = + - 7a b + Property () Formulas (7), (), (5) (d) Notice that the idex of summatio starts at. We use property () as follows: 0 a k 0 = a k 0 = B a k 5 - a k R = B 0 k = k = æ æ æ Property () Property () Formula (7) = =,0 Now Work PROBLEM 7-5 R. Assess Your Uderstadig Are You Prepared? Aswers are g ive at the ed of these ex ercises. If y ou g et a wrog aswer, read the pag es listed i red.. For the fuctio fx = x -, fid f ad f.. True or False A fuctio is a relatio betwee two sets D x (pp. 5) ad R so that each elemet x i the first set D is related to exactly oe elemet y i the secod set R. (pp. 5)

8 800 CHAPTER Sequeces; Iductio; the Biomial Theorem Cocepts ad Vocabulary. A() is a fuctio whose domai is the set of positive itegers.. True or False The otatio represets the fifth term of a sequece. 5. If Ú 0 is a iteger, the! = whe Ú.. The sequece a = 5, a = a - is a example of a sequece. a 5 7. The otatio a + a + a + Á + a = a is a example of otatio. 8. True or False + a Á + = a k Skill Buildig I Problems 9, evaluate each factorial ex pressio. 9. 0! 0. 9!. 9!!.! 0!.! 7!!. 5! 8!! I Problems 5, write dow the first five terms of each sequece. 5. {s } = 5. {s } = {a } = e + f 8. {b } = e + f 9. {c } = {d } = e - - a - bf. {s } = b + r. {s } = ea b f. {t } = b r. {a } = b r 5. {b } = b e r. {c } = b r I Problems 7, the g ive patter cotiues. Write dow the th term of a sequece {a } sug g ested by the patter. 7.,,, 5, Á 8. #, Á #, #, #, 9., 5,, 8, Á 0. 8, 9, 7, 8, Á., -,, -,, -, Á.,,, 5, 7,,, 8, Á., -,, -, 5, -, Á., -,, - 8, 0, Á I Problems 5 8, a sequece is defied recursively. Write dow the first five terms. 5. a = ; a = + a -. a = ; a = - a - 7. a = - ; a = + a - 8. a = ; a = - a - 9. a = 5; a = a - 0. a = ; a = - a -. a = ; a = a -. a = - ; a = + a -. a = ; a = ; a = a - # a -. a =- ; a = ; a = a - + a - 5. a = A; a = a - + d. a = A; a = ra -, r Z 0 7. a = ; a = + a - 8. a = ; a = A a - I Problems 9 58, write out each sum. 9. a 5. a a k b k a a k k + 5. a I Problems 59 8, ex press each sum usig summatio otatio a k + 5. a 5. k + a k a - k l k 58. a - k + k k = k = Á Á Á Á Á Á + - a b k

9 SECTION. Sequeces Á +. e + e + e + Á + e 7. a + a + d + a + d + Á + a + d 8. a + ar + ar + Á + ar - I Problems 9 80, fid the sum of each sequece a a Applicatios ad Extesios 0 7. a 5k + 7. a a k a k 78. a - k 79. a 0 k = 8 k = a - k k a k + 7. a k - 0 k 80. a k k = 8. Credit Card Debt Joh has a balace of $000 o his Discover card that charges % iterest per moth o ay upaid balace. Joh ca afford to pay $00 toward the balace each moth. His balace each moth after makig a $00 paymet is give by the recursively defied sequece B 0 = $000 B =.0B Determie Joh s balace after makig the first paymet. That is, determie B. 8. Trout Populatio A pod curretly has 000 trout i it. A fish hatchery decides to add a additioal 0 trout each moth. I additio, it is kow that the trout populatio is growig % per moth. The size of the populatio after moths is give by the recursively defied sequece p 0 = 000 p =.0p How may trout are i the pod after two moths? That is, what is p? 8. Car Loas Phil bought a car by takig out a loa for $8,500 at 0.5% iterest per moth. Phil s ormal mothly paymet is $.7 per moth, but he decides that he ca afford to pay $00 extra toward the balace each moth. His balace each moth is give by the recursively defied sequece B 0 = $8,500 B =.005B Determie Phil s balace after makig the first paymet. That is, determie B. 8. Evirometal Cotrol The Evirometal Protectio Agecy (EPA) determies that Maple Lake has 50 tos of pollutat as a result of idustrial waste ad that 0% of the pollutat preset is eutralized by solar oxidatio every year. The EPA imposes ew pollutio cotrol laws that result i 5 tos of ew pollutat eterig the lake each year. The amout of pollutat i the lake after years is give by the recursively defied sequece p 0 = 50 p = 0.9p Determie the amout of pollutat i the lake after years. That is, determie p. 85. Growth of a Rabbit Coloy A coloy of rabbits begis with oe pair of mature rabbits, which will produce a pair of offsprig (oe male, oe female) each moth. Assume that all rabbits mature i moth ad produce a pair of offsprig (oe male, oe female) after moths. If o rabbits ever die, how may pairs of mature rabbits are there after 7 moths? [Hit: A Fiboacci sequece models this coloy. Do you see why?] mature pair mature pair mature pairs mature pairs 8. Fiboacci Sequece Let u = A + 5B - A - 5B 5 defie the th term of a sequece. (a) Show that u = ad u =. (b) Show that u + = u + + u. (c) Draw the coclusio that 5u is a Fiboacci sequece. 87. Pascal s Triagle Divide the triagular array show (called Pascal s triagle) usig diagoal lies as idicated. Fid the sum of the umbers i each diagoal row. Do you recogize this sequece? Fiboacci Sequece Use the result of Problem 8 to do the followig problems: (a) Write the first terms of the Fiboacci sequece. u + u (b) Write the first 0 terms of the ratio. (c) As gets large, what umber does the ratio approach? This umber is referred to as the golde ratio. Rectagles whose sides are i this ratio were cosidered pleasig to

10 80 CHAPTER Sequeces; Iductio; the Biomial Theorem the eye by the Greeks. For example, the façade of the Partheo was costructed usig the golde ratio. (d) Write dow the first 0 terms of the ratio. (e) As gets large, what umber does the ratio approach? This umber is referred to as the cojugate golde ratio. This ratio is believed to have bee used i the costructio of the Great Pyramid i Egypt. The ratio equals the sum of the areas of the four face triagles divided by the total surface area of the Great Pyramid. 89. Approximatig f(x) e x I calculus, it ca be show that fx = e x = a q We ca approximate the value of fx = e x for ay x usig the followig sum fx = e x L a for some. (a) Approximate f(.) with = (b) Approximate f(.) with = 7. (c) Use a calculator to approximate f(.). (d) Usig trial ad error alog with a graphig utility s SEQuece mode, determie the value of required to approximate f(.) correct to eight decimal places. 90. Approximatig f(x) e x Refer to Problem 89. (a) Approximate f-. with =. (b) Approximate f-. with =. (c) Use a calculator to approximate f-.. (d) Usig trial ad error alog with a graphig utility s SEQuece mode, determie the value of required to approximate f-. correct to eight decimal places. 9. Bode s Law I 77, Joha Bode published the followig formula for predictig the mea distaces, i astroomical uits (AU), of the plaets from the su: a = 0. a = # -, Ú where is the umber of the plaet from the su. x k k! x k k! u u + (a) Determie the first eight terms of this sequece. (b) At the time of Bode s publicatio, the kow plaets were Mercury (0.9 AU), Veus (0.7 AU), Earth ( AU), Mars (.5 AU), Jupiter (5.0 AU), ad Satur (9.5 AU). How do the actual distaces compare to the terms of the sequece? (c) The plaet Uraus was discovered i 78 ad the asteroid Ceres was discovered i 80. The mea orbital distaces from the su to Uraus ad Ceres* are 9. AU ad.77 AU, respectively. How well do these values fit withi the sequece? (d) Determie the ith ad teth terms of Bode s sequece. (e) The plaets Neptue ad Pluto* were discovered i 8 ad 90, respectively. Their mea orbital distaces from the su are 0.07 AU ad 9. AU, respectively. How do these actual distaces compare to the terms of the sequece? (f) O July 9, 005, NASA aouced the discovery of a dwarf plaet* ( = ), which has bee amed Eris. Use Bode s Law to predict the mea orbital distace of Eris from the su. Its actual mea distace is ot yet kow, but Eris is curretly about 97 astroomical uits from the su. Source: NASA. 9. Show that [Hit: Let Á = S = + + Á S = Á + Add these equatios. The S [ ] [ ( )]... [ ] terms i brackets Now complete the derivatio.] * Ceres, Haumea, Makemake, Pluto, ad Eris are referred to as dwarf plaets. Computig Square Roots A method for approx imatig p ca 97. Triagular Numbers A triagular umber is a term of the be traced back to the Baby loias. The formula is g ive by the sequece recursively defied sequece u =, u + = u + + a 0 = k a = aa p - + b Write dow the first seve triagular umbers. a For the sequece give i Problem 97, show that where k is a iitial g uess as to the value of the square root. Use this recursive formula to approx imate the followig square roots by fidig + + u + =. a 5. Compare this result to the value provided by y our calculator Explaiig Cocepts: Discussio ad Writig 99. For the sequece give i Problem 97, show that u + + u = Ivestigate various applicatios that lead to a Fiboacci sequece, such as art, architecture, or fiacial markets. Write a essay o these applicatios. 0. Write a paragraph that explais why the umbers foud i Problem 97 are called triagular. Are You Prepared? Aswers. f =. True ; f =