Sequences and Series

Transcription

1 CHAPTER Uit 4 Probability, Data Aalysis, ad Discrete Math Sequeces ad Series Ifiite Series, p (0.7) Prerequisite Skills Defie ad Use Sequeces ad Series Graphig Calculator Activity Work with Sequeces Aalyze Arithmetic Sequeces ad Series Aalyze Geometric Sequeces ad Series Mixed Review of Problem Solvig Fid Sums of Ifiite Geometric Series Ivestigatig Algebra: Ivestigatig a Ifiite Geometric Series Use Recursive Rules with Sequeces ad Fuctios Ivestigatig Algebra: Explorig Recursive Rules Problem Solvig Workshop Mixed Review of Problem Solvig ASSESSMENT Quizzes... 87, 833 Chapter Summary ad Review Chapter Test Stadardized Test Preparatio ad Practice Activities , 80, 8, 80, 83 Chapter Highlights PROBLEM SOLVING Mixed Review of Problem Solvig, 88, 838 Multiple Represetatios, 808, 86 Multi-Step Problems, 800, 88, 838 Usig Alterative Methods, 834 Real-World Problem Solvig Examples, 79, 80, 83, 8, 89 ASSESSMENT Stadardized Test Practice Examples, 80, 8 Multiple Choice, 798, 799, 806, 807, 84, 8, 83, 830, 83, 844 Short Respose/Exteded Respose, 799, 800, 809, 86, 88, 84, 8, 83 Writig/Ope-Eded, 798, 806, 807, 84, 8, 88, 83, 84, 830, 83 TECHNOLOGY At classzoe.com: Aimated Algebra, 793, 80, 8, 80, Tutor, 79, 799, 80, 808, 8, 84, 86, 83, 840 Olie Quiz, 800, 809, 87, 8, 833 State Test Practice, 76, 838, 847 Cotets xix

2 Sequeces ad Series. Defie ad Use Sequeces ad Series. Aalyze Arithmetic Sequeces ad Series.3 Aalyze Geometric Sequeces ad Series.4 Fid Sums of Ifiite Geometric Series. Use Recursive Rules with Sequeces ad Fuctios Before I previous chapters, you leared the followig skills, which you ll use i Chapter : solvig equatios, solvig systems of equatios, ad performig fuctio compositio. Prerequisite Skills VOCABULARY CHECK ad g(x) 4x. Copy ad complete the statemet usig f (x) x. The domai of f(x) is?.. The rage of g(x) is?. 3. The compositio f (g(x)) is equal to?. SKILLS CHECK Solve the equatio. Check your solutio. (Review p. 8 for..) 4. 7x x x x 8 8. x 9 3x 7 9. x 3 6 x Solve the system usig ay algebraic method. (Review p. 60 for.3.) 0. 3x y 0. x y 0 x 4y 30 x y 0. 4x y 0.x.y 8. Let f(x) x ad g(x) x. Perform the idicated operatio ad state the domai. (Review p. 48 for..) 3. f(g(x)) 4. f(f (x)). g(g(x)) SFSFRVJTJUF TLJMMT QSBDUJDF BU DMBTT[POF DPN 79 pe-00.idd 79 0/6/0 :9:3 AM

3 Now I Chapter, you will apply the big ideas listed below ad reviewed i the Chapter Summary o page 839. You will also use the key vocabulary listed below. Big Ideas Aalyze sequeces Fid sums of series 3 Use recursive rules KEY VOCABULARY sequece, p. 794 arithmetic sequece, p. 80 geometric series, p. 8 terms of a sequece, p. 794 commo differece, p. 80 partial sum, p. 80 series, p. 796 arithmetic series, p. 804 explicit rule, p. 87 summatio otatio, p. 796 geometric sequece, p. 80 recursive rule, p. 87 sigmotatio, p. 796 commo ratio, p. 80 iteratio, p. 830 Why? You ca use sequeces to describe patters i the real world. For example, you ca use the Fiboacci sequece to describe patters i ature. Algebra The aimatio illustrated below for Example 3 o page 88 helps you aswer this questio: How ca you geerate Fiboacci umbers? A A AN AN AN A A A A A #HECK!NSWER 3TART Fiboacci umbers are see i objects such as shells, piecoes, ad broccoli. Use the recursive rule to fid umbers i the Fiboacci sequece. Algebra at classzoe.com Other aimatios for Chapter : pages 80, 8, ad pe-00.idd 793 0/6/0 :00: PM

4 . Defie ad Use Sequeces ad Series Before You idetified ad wrote fuctios. Now You will recogize ad write rules for umber patters. Why? So you ca fid agle measures, as i Ex. 63. Key Vocabulary sequece terms of a sequece series summatio otatio sigmotatio KEY CONCEPT Sequeces For Your Notebook A sequece is a fuctio whose domai is a set of cosecutive itegers. If a domai is ot specified, it is uderstood that the domai starts with. The values i the rage are called the terms of the sequece. Domai: The relative positio of each term Rage: a a a 3 a 4... Terms of the sequece A fiite sequece has a limited umber of terms. A ifiite sequece cotiues without stoppig. Fiite sequece:, 4, 6, 8 Ifiite sequece:, 4, 6, 8,... A sequece ca be specified by a equatio, or rule. For example, both sequeces above ca be described by the rule or f(). E XAMPLE Write terms of sequeces Write the first six terms of (a) ad (b) f() (3). Solutio a. a () 7 st term b. f() (3) st term a () 9 d term f() (3) 3 d term a 3 (3) 3rd term f(3) (3) 3 9 3rd term a 4 (4) 3 4th term f(4) (3) 4 7 4th term a () th term f() (3) 8 th term a 6 (6) 7 6th term f(6) (3) th term GUIDED PRACTICE for Example Write the first six terms of the sequece.. 4. f() () Chapter Sequeces ad Series

5 WRITING RULES If the terms of a sequece have a recogizable patter, the you may be able to write a rule for the th term of the sequece. E XAMPLE Write rules for sequeces WRITE RULES If you are give oly the first several terms of a sequece, there is o sigle rule for the th term. For istace, the sequece, 4, 8,... ca be give by or. Describe the patter, write the ext term, ad write a rule for the th term of the sequece (a), 8, 7, 64,... ad (b) 0,, 6,,.... Solutio a. You ca write the terms as () 3, () 3, (3) 3, (4) 3,.... The ext term is a () 3. A rule for the th term is () 3. b. You ca write the terms as 0(), (), (3), 3(4),.... The ext term is f() 4() 0. A rule for the th term is f() ( ). GRAPHING SEQUENCES To graph a sequece, let the horizotal axis represet the positio umbers (the domai) ad the vertical axis represet the terms (the rage). E XAMPLE 3 Solve a multi-step problem RETAIL DISPLAYS You work i a grocery store ad are stackig apples i the shape of a square pyramid with 7 layers. Write a rule for the umber of apples i each layer. The graph the sequece. Solutio STEP Make a table showig the umber of fruit i the first three layers. Let represet the umber of apples i layer. Layer, 3 Number of apples, AVOID ERRORS Although the plotted poits i Example 3 follow a curve, do ot draw the curve because the sequece is defied oly for iteger values of. STEP Write a rule for the umber of apples i each layer. From the table, you ca see that. STEP 3 Plot the poits (, ), (, 4), (3, 9),..., (7, 49). The graph is show at the right. Number of apples Layer GUIDED PRACTICE for Examples ad 3 4. For the sequece 3, 8,, 4,..., describe the patter, write the ext term, graph the first five terms, ad write a rule for the th term.. WHAT IF? I Example 3, suppose there are 9 layers of apples. How may apples are i the 9th layer?. Defie ad Use Sequeces ad Series 79

6 KEY CONCEPT For Your Notebook Series ad Summatio Notatio Whe the terms of a sequece are added together, the resultig expressio is a series. A series ca be fiite or ifiite. Fiite series: Ifiite series: You ca use summatio otatio to write a series. For example, the two series above ca be writte i summatio otatio as follows: READING Whe writte i summatio otatio, this series is read as the sum of i for values of i from to i For both series, the idex of summatio is i ad the lower limit of summatio is. The upper limit of summatio is 4 for the fiite series ad (ifiity) for the ifiite series. Summatio otatio is also called sigmotatio because it uses the uppercase Greek letter sigma, writte S. i E XAMPLE 4 Write series usig summatio otatio Write the series usig summatio otatio. a b. Solutio a. Notice that the first term is (), the secod is (), the third is (3), ad the last is (0). So, the terms of the series ca be writte as: a i i where,, 3,..., 0 The lower limit of summatio is ad the upper limit of summatio is 0. c The summatio otatio for the series is 0 i. b. Notice that for each term the deomiator of the fractio is more tha the umerator. So, the terms of the series ca be writte as: a i i where,, 3, 4,... i The lower limit of summatio is ad the upper limit of summatio is ifiity. i c The summatio otatio for the series is i. GUIDED PRACTICE for Example 4 Write the series usig summatio otatio Chapter Sequeces ad Series

7 INDEX OF SUMMATION The idex of summatio for a series does ot have to be i ay letter ca be used. Also, the idex does ot have to begi at. For istace, the idex begis at 4 i the ext example. AVOID ERRORS Be sure to use the correct lower ad upper limits of summatio whe fidig the sum of a series. E XAMPLE Fid the sum of the series. Fid the sum of a series 8 (3 k ) (3 4 ) (3 ) (3 6 ) (3 7 ) (3 8 ) k SPECIAL FORMULAS For series with may terms, fidig the sum by addig the terms ca be tedious. Below are formulas you ca use to fid the sums of three special types of series. KEY CONCEPT For Your Notebook Formulas for Special Series Sum of terms of Sum of first positive itegers Sum of squares of first positive itegers i ( ) i ( )( ) 6 E XAMPLE 6 Use a formula for a sum RETAIL DISPLAYS How may apples are i the stack i Example 3 o page 79? Solutio From Example 3 you kow that the ith term of the series is give by a i i where,, 3,..., 7. Usig summatio otatio ad the third formula listed above, you ca fid the total umber of apples as follows: i 7(7 )( p 7 ) 7(8)() c There are 40 apples i the stack. Check this by actually addig the umber of apples i each of the seve layers. GUIDED PRACTICE for Examples ad 6 Fid the sum of the series. 0. 8i. 7 k 3 (k ) WHAT IF? Suppose there are 9 layers i the apple stack i Example 3. How may apples are i the stack?. Defie ad Use Sequeces ad Series 797

8 . EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS for Exs. 9, 47, ad 6 STANDARDIZED TEST PRACTICE Exs., 7, 8, 64, ad 67. VOCABULARY Copy ad complete: Aother ame for summatio otatio is?.. WRITING Explai the differece betwee a sequece ad a series. EXAMPLE o p. 794 for Exs. 3 4 WRITING TERMS Write the first six terms of the sequece f() f() 0. ( 3). f() f() EXAMPLE o p. 79 for Exs. 7 WRITING RULES For the sequece, describe the patter, write the ext term, ad write a rule for the th term.., 6,, 6,... 6.,, 4, 8, , 8,, 6,... 8., 9, 8, 6, , 6, 9,, , 4 4, 6, 8 6,... 4, 4, 3 4, 4 4, 4,.... 0, 3 0, 30, 7, , 3.8, 4.,., ,.6,, 0.6,.,....., 4., 9., 6., , 6.8, 4.6, 3.4, MULTIPLE CHOICE Which rule gives the total umber of squares i the th figure of the patter show? 3 4 A 3 3 B 4 C D ( ) EXAMPLE 3 o p. 79 for Exs GRAPHING SEQUENCES Graph the sequece. 8.,, 8,, 4 9., 4, 8, 6, 3, ,, 9, 3,..., 9 3., 4, 6, 8,..., 3. 0, 3, 8,, 4, 3 33., 0,, 8, , 9, 4, 9, 4 3., 3,,..., , 8, 3 7, 4 6,..., 9 EXAMPLE 4 o p. 796 for Exs WRITING SUMMATION NOTATION Write the series usig summatio otatio Chapter Sequeces ad Series

9 EXAMPLES ad 6 o p. 797 for Exs. 4 8 USING SUMMATION NOTATION Fid the sum of the series k 3 i 46. (k ) 0. 7i 47. ( ) i. 4 k 6 k 3k k k i ERROR ANALYSIS Describe ad correct the error i fidig the sum of the series. i 0 (i 3) MULTIPLE CHOICE What is the sum of the series 0 A 0 B 0 C 40 D 870 i? REVIEW LOGIC For help with couterexamples see p. 00. CHALLENGE Tell whether the statemet about summatio otatio is true or false. If the statemet is true, prove it. If the statemet is false, give a couterexample ka i k a i b i a i 60. a i b i 6. ( a i b i ) ( a i ) k k a i a i b i PROBLEM SOLVING EXAMPLES 3 ad 6 o pp for Exs GEOMETRY For a regular -sided polygo ( 3), the measure of a iterior agle is give by this formula: 80( ) Write the first five terms of the sequece. Write a rule for the sequece givig the total measure T of the iterior agles i each regular -sided polygo. Use the rule to fid the total measure of the agles i the Guggeheim Museum skylight, which is a regular dodecago. Guggeheim Museum Skylight 64. SHORT RESPONSE You wat to save $00 for a school trip. You begi by savig a pey o the first day. You pla to save a additioal pey each day after that. For example, you will save peies o the secod day, 3 peies o the third day, ad so o. How much moey will you have saved after 00 days? How may days must you save to have saved $00? Explai how you used a series to fid your aswer.. Defie ad Use Sequeces ad Series 799

10 6. TOWER OF HANOI I the puzzle called the Tower of Haoi, the object is to use a series of moves to take the rigs from oe peg ad stack them i order o aother peg. A move cosists of movig exactly oe rig, ad o rig may be placed o top of a smaller rig. The miimum umber of moves required to move rigs is for rig, 3 for rigs, 7 for 3 rigs, for 4 rigs, ad 3 for rigs. Fid a formula for the sequece. What is the miimum umber of moves required to move 6 rigs? 7 rigs? 8 rigs? Start Step Step Step 3 Ed 66. MULTI-STEP PROBLEM The mea distace d (i astroomical uits) of each plaet (except Neptue) from the su is approximated by the Titius-Bode rule, d 0.3() 0.4, where is a positive iteger represetig the positio of the plaet from the su. a. Evaluate The value of is 4 for Mars. Use the Titius-Bode rule to approximate the distace of Mars from the su. b. Covert Oe astroomical uit is equal to about 49,600,000 kilometers. How far is Mars from the su i kilometers? c. Graph Graph the sequece give by the Titius-Bode rule. 67. EXTENDED RESPONSE For a display at a sports store, you are stackig soccer balls i a pyramid whose base is a equilateral triagle. The ( ) umber of balls per layer is give by where represets the top layer. a. How may balls are i the fifth layer? b. How may balls are i a stack with five layers? c. Compare the umber of balls i a layer of a triagular pyramid with the umber of balls i the same layer of a square pyramid. 68. CHALLENGE Usig the true statemets from Exercises 9 6 o page 799 ad the special formulas o page 797, fid a formula for the umber of balls i the top layers of the pyramid from Exercise 67. MIXED REVIEW PREVIEW Prepare for Lesso. i Exs Solve the equatio. Check your solutio. (p. 8) x x 7. x 7. 8x x x x x 77. x 6 39 Fid the distace betwee the poits. (p. 64) 78. (4, ), (6, ) 79. (7, 4), (, ) 80. (0, ), (, ) 8. (4, 6), (, 9) 8. (, ), (6, 4) 83. (, 4), (, 8) 84. (9, 7), (, 6) 8. (, 8), (3, ) 86. (4, 0), (9, 6) 800 Chapter EXTRA Sequeces PRACTICE ad Series for Lesso., p. 0 ONLINE QUIZ at classzoe.com

11 . Work with Sequeces Use after Lesso. classzoe.com Keystrokes QUESTION How ca you use a graphig calculator to perform operatios with sequeces? EXAMPLE Fid, graph, ad sum terms of a sequece Use a graphig calculator to fid the first eight terms of 3. Graph the sequece. The fid the sum of the first eight terms of the sequece. STEP Eter sequece Put the graphig calculator i sequece mode ad dot mode. Eter the sequece. Note that the calculator uses u() rather tha. STEP Calculate terms Use the table feature to view the terms of the sequece. The first eight terms are, 7,, 7,, 7, 3, ad 37. Mi= u()=-3 u(mi)= v()= v(mi)= w()= w(mi)= 3 4 = u() 7 7 STEP 3 Graph sequece Set the viewig widow so that 8, 0 x 9, ad 0 y 40. Graph the sequece. Use the trace feature to view the terms of the sequece. STEP 4 Fid sum of terms Use the summatio feature to fid the sum of the first eight terms of the sequece. The scree shows that the sum is 6. sum(seq(-3,,, 8)) 6 = X= Y= P RACTICE Use a graphig calculator to (a) fid the first te terms of the sequece, (b) graph the sequece, ad (c) fid the sum of the first te terms of the sequece ( )

12 . Aalyze Arithmetic Sequeces ad Series Before You worked with geeral sequeces ad series. Now You will study arithmetic sequeces ad series. Why? So you ca arrage a marchig bad, as i Ex. 64. Key Vocabulary arithmetic sequece commo differece arithmetic series I a arithmetic sequece, the differece of cosecutive terms is costat. This costat differece is called the commo differece ad is deoted by d. E XAMPLE Idetify arithmetic sequeces Tell whether the sequece is arithmetic. a. 4,, 6,, 6,... b. 3,, 9,, 3,... Solutio Fid the differeces of cosecutive terms. a. a a (4) b. a a 3 a 3 a 6 a 3 a 9 4 a 4 a 3 6 a 4 a a a 4 6 a a c Each differece is, so the sequece is arithmetic. c The differeces are ot costat, so the sequece is ot arithmetic. GUIDED PRACTICE for Example. Tell whether the sequece 7, 4,, 8,,... is arithmetic. Explai why or why ot. KEY CONCEPT For Your Notebook Rule for a Arithmetic Sequece Algebra Example The th term of a arithmetic sequece with first term a ad commo differece d is give by: a ( )d The th term of a arithmetic sequece with a first term of ad commo differece 3 is give by: ( )3, or 3 80 Chapter Sequeces ad Series

13 E XAMPLE Write a rule for the th term Write a rule for the th term of the sequece. The fid a. a. 4, 9, 4, 9,... b. 60,, 44, 36,... Solutio AVOID ERRORS I the geeral rule for a arithmetic sequece, ote that the commo differece d is multiplied by, ot. a. The sequece is arithmetic with first term a 4 ad commo differece d 9 4. So, a rule for the th term is: a ( )d Write geeral rule. 4 ( ) Substitute 4 for a ad for d. Simplify. The th term is a () 74. b. The sequece is arithmetic with first term a 60 ad commo differece d So, a rule for the th term is: a ( )d Write geeral rule. 60 ( )(8) Substitute 60 for a ad 8 for d Simplify. The th term is a 68 8(). E XAMPLE 3 Write a rule give a term ad commo differece Oe term of a arithmetic sequece is a The commo differece is d 3. a. Write a rule for the th term. b. Graph the sequece. Solutio a. Use the geeral rule to fid the first term. a ( )d Write geeral rule. a 9 a (9 )d Substitute 9 for. 48 a 8(3) Substitute 48 for a 9 ad 3 for d. 6 a Solve for a. So, a rule for the th term is: a ( )d Write geeral rule. 6 ( )3 Substitute 6 for a ad 3 for d. 9 3 Simplify. b. Create a table of values for the sequece. The graph of the first 6 terms of the sequece is show. Notice that the poits lie o a lie. This is true for ay arithmetic sequece Aalyze Arithmetic Sequeces ad Series 803

14 E XAMPLE 4 Write a rule give two terms Two terms of a arithmetic sequece are a 8 ad a Fid a rule for the th term. Solutio STEP Write a system of equatios usig a ( )d ad substitutig 7 for (Equatio ) ad the 8 for (Equatio ). a 7 a (7 )d 97 a 6d Equatio a 8 a (8 )d a 7d Equatio STEP Solve the system. 76 9d Subtract. 4 d Solve for d. 97 a 6(4) Substitute for d i Equatio. 7 a Solve for a. STEP 3 Fid a rule for. a ( )d Write geeral rule. 7 ( )4 Substitute for a ad d. 4 Simplify. GUIDED PRACTICE for Examples, 3, ad 4 Write a rule for the th term of the arithmetic sequece. The fid a 0.. 7, 4,, 8, a 7, d 7 4. a 7 6, a 6 7 ARITHMETIC SERIES The expressio formed by addig the terms of a arithmetic sequece is called a arithmetic series. The sum of the first terms of a arithmetic series is deoted by S. To fid a rule for S, you ca write S i two differet ways ad add the results. S a (a d) (a d)... S ( d) ( d)... a S (a ) (a ) (a )... (a ) You ca coclude that S (a ), which leads to the followig result. KEY CONCEPT For Your Notebook The Sum of a Fiite Arithmetic Series The sum of the first terms of a arithmetic series is: S a I words, S is the mea of the first ad th terms, multiplied by the umber of terms. 804 Chapter Sequeces ad Series

15 E XAMPLE Stadardized Test Practice What is the sum of the arithmetic series 0 (3 i)? A 03 B C 0 D 0 CLASSIFY SERIES You ca verify that the series i Example is arithmetic by evaluatig 3 i for the first few values of the idex i. The resultig terms are 8, 3, 8, 3,..., which have a commo differece of. Solutio a 3 () 8 a 0 3 (0) 03 Idetify first term. Idetify last term. S Write rule for S 0, substitutig 8 for a ad 03 for a 0. 0 Simplify. c The correct aswer is C. A B C D E XAMPLE 6 Use a arithmetic sequece ad series i real life HOUSE OF CARDS You are makig a house of cards similar to the oe show. a. Write a rule for the umber of cards i the th row if the top row is row. b. What is the total umber of cards if the house of cards has 4 rows? first row Solutio a. Startig with the top row, the umbers of cards i the rows are 3, 6, 9,,.... These umbers form a arithmetic sequece with a first term of 3 ad a commo differece of 3. So, a rule for the sequece is: a ( )d Write geeral rule. 3 ( )3 Substitute 3 for a ad 3 for d. 3 Simplify. b. Fid the sum of a arithmetic series with first term a 3 ad last term a 4 3(4) 4. Total umber of cards S 4 4 a a 4 at classzoe.com GUIDED PRACTICE for Examples ad 6. Fid the sum of the arithmetic series ( 7i). 6. WHAT IF? I Example 6, what is the total umber of cards if the house of cards has 8 rows?. Aalyze Arithmetic Sequeces ad Series 80

16 . EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS for Exs., 4, ad 6 STANDARDIZED TEST PRACTICE Exs., 9, 39,, ad 68 MULTIPLE REPRESENTATIONS Ex. 66. VOCABULARY Copy ad complete: The costat differece betwee cosecutive terms of a arithmetic sequece is called the?.. WRITING Explai the differece betwee a arithmetic sequece ad a arithmetic series. EXAMPLE o p. 80 for Exs. 3 IDENTIFYING ARITHMETIC SEQUENCES Tell whether the sequece is arithmetic. Explai why or why ot. 3.,,, 8,, , 4,, 6, 3,...., 4, 3, 3, 4, , 7,,, 0, ,,.,,., , 0,,.,., , 4, 3 4, 3 4, 4, , 7, 4 7, 8 7, 6 7,....,,,, 7,... EXAMPLE o p. 803 for Exs. WRITING RULES Write a rule for the th term of the arithmetic sequece. The fid a 0.., 4, 7, 0, 3,... 3.,, 7, 3, 9, ,, 34, 47, 60,.... 3,,, 3,, ,,, 6, 0,... 7., 4, 3, 8, 9, , 3, 4 3,, 8 3,... 9., 3, 4 3,,, , 3.6,.7, 7.8, 9.9,... 3 ERROR ANALYSIS Describe ad correct the error i writig the rule for the th term of the arithmetic sequece 37, 4,,,,..... Use a 37 ad d 3. a d 37 (3) The first term is 37 ad the commo differece is 3. 3 ( )(37) 0 37 EXAMPLE 3 o p. 803 for Exs. 3 9 WRITING RULES Write a rule for the th term of the arithmetic sequece. The graph the first six terms of the sequece. 3. a 6, d 4. a 6 6, d 9. a 4 96, d 4 6. a 3, d 7 7. a 0 30, d 7 8. a, d 9. MULTIPLE CHOICE For a certai arithmetic sequece, a 30 7 ad d 4. What is a rule for the th term of the sequece? A 63 4 C 63 4 B 9 4 D Chapter Sequeces ad Series

17 EXAMPLE 4 o p. 804 for Exs WRITING RULES Write a rule for the th term of the arithmetic sequece that has the two give terms. 30. a 4 3, a a 6 39, a a 3, a a 8 0, a a 9 89, a a 7, a a 7 4, a a, a a 6 0, a 39. MULTIPLE CHOICE For a certai arithmetic sequece, a 6 6 ad a What is a rule for the th term of the sequece? A 8 6 C 6 4 B 30 6 D 36 6 EXAMPLE o p. 80 for Exs FINDING SUMS Fid the sum of the arithmetic series ( 3i) 4. (9 i) i 3 (3 i) 4. (7 6i) i (4 6i) (4 9i) USING GRAPHS Write a rule for the sequece whose graph is show (4, 7) (3, ) (, 7) (, ) 0. (, ) (, 4) (3, 7) (4, 0). (, 3) (, ) (3, 7) (4, 9). WRITING Compare the graph of 3, where is a positive iteger, with the graph of f(x) 3x, where x is a real umber. Discuss how the graph of a arithmetic sequece is similar to ad differet from the graph of a liear fuctio. REASONING Tell whether the statemet is true or false. Explai your aswer. 3. If the commo differece of a arithmetic series is doubled while the first term ad umber of terms i the series remai uchaged, the the sum of the series is doubled. 4. If the umbers a, b, ad c are the first three terms of a arithmetic sequece, the b is half the sum of a ad c. SOLVING EQUATIONS Fid the value of.. 8. ( 7i) ( i) 0 9. i 3 (0 3i) 8 7. (3 4i) i (8 8i) 0 (7 i) 4 6. REASONING Fid the sum of all positive odd itegers less tha CHALLENGE The umbers 3 x, x, ad 3x are the first three terms i a arithmetic sequece. Fid the value of x ad the ext term i the sequece.. Aalyze Arithmetic Sequeces ad Series 807

18 PROBLEM SOLVING EXAMPLE 6 o p. 80 for Exs HONEYCOMBS Domestic bees make their hoeycomb by startig with a sigle hexagoal cell, the formig rig after rig of hexagoal cells aroud the iitial cell, as show. The umbers of cells i successive rigs form a arithmetic sequece. a. Write a rule for the umber of cells i the th rig. b. What is the total umber of cells i the hoeycomb after the 9th rig is formed? (Hit: Do ot forget to cout the iitial cell.) Iitial cell rig rigs 64. MARCHING BAND A marchig bad is arraged i 7 rows. The first row has 3 bad members, ad each row after the first has more bad members tha the row before it. Write a rule for the umber of bad members i the th row. The fid the total umber of bad members. 6. SCULPTURE Sol LeWitt s sculpture Four-Sided Pyramid i the Natioal Gallery of Art Sculpture Garde is made of cocrete blocks. As show i the diagram, each layer has 8 more visible blocks tha the layer i frot of it. a. Write a rule for the umber of visible blocks i the th layer where represets the frot layer. b. Whe you view the pyramid from oe corer, a total of layers are visible. How may of the pyramid s blocks are visible? 66. MULTIPLE REPRESENTATIONS The distace D (i feet) that a object falls i t secods ca be modeled by D(t) 6t. a. Makig a Table Let d() represet the distace the object falls i the th secod. Make a table of values showig d(), d(), d(3), ad d(4). (Hit: The distace d() that the object falls i the first secod is D() D(0).) b. Writig a Rule Write a rule for the sequece of distaces give by d(). c. Drawig a Graph Graph the sequece from part (b). 67. ENTERTAINMENT Durig a high school spirit week, studets dress up i costumes. A cash prize is give each day to the studet with the best costume. The orgaizig committee has $000 to give away over five days. The committee wats to icrease the amout of the prize by $0 each day. How much should the committee give away o the first day? WORKED-OUT SOLUTIONS 808 Chapter Sequeces o p. WS ad Series STANDARDIZED TEST PRACTICE MULTIPLE REPRESENTATIONS

19 68. EXTENDED RESPONSE A paper towel maufacturer sells paper towels rolled oto cardboard dowels. The thickess of the paper is ich. The diameter of a dowel is iches, ad the total diameter of a roll is iches. d (i.) l (i.) π?? 3?? 4??. i. i.. i. i. a. Calculate Let be the umber of times the paper towel is wrapped aroud the dowel, let d be the diameter of the roll just before the th wrap, ad let l be the legth of paper added i the th wrap. Copy ad complete the table. b. Model What kid of sequece is l, l, l 3, l 4,...? Write a rule for the th term of the sequece. c. Apply Fid the umber of times the paper must be wrapped aroud the dowel to create a roll with a ich diameter. Use your aswer ad the rule from part (b) to fid the legth of paper i a roll with a ich diameter. d. Iterpret Suppose a roll with a ich diameter costs $.0. How much would you expect to pay for a roll with a 7 ich diameter whose dowel also has a diameter of iches? Explai your reasoig ad ay assumptios you make. 69. CHALLENGE A theater has rows of seats, ad each row has d more seats tha the row i frot of it. There are x seats i the last (th) row ad a total of y seats i the etire theater. How may seats are i the frot row of the theater? Write your aswer i terms of, x, ad y. MIXED REVIEW PREVIEW Prepare for Lesso.3 i Exs Solve the equatio. 70. x / 7 (p. 4) 7. x /3 36 (p. 4) 7. 6x / 9 (p. 4) 73. 3x 3/ 4 (p. 4) 74. (x 0) /4 (p. 4) 7. (x 3) 3/4 64 (p. 4) x 6 (p. ) 77. x 3 (p. ) x 6 (p. ) x 9 (p. ) x 3 49 x 8 (p. ) x 79 x (p. ) Fid the mea, media, ad mode of the data set. (p. 744) 8., 6, 6, 6, 8, , 36, 38, 43, 43, 4, , 8, 9, 80, 7, 83, 9, , 4,,,,, ,.9,.6,.9,.,.8, , 6.7, 3.8, 4.,.,.8, CRAFT FAIR You are sellig hadmade hats ad scarves at a craft fair. You charge $6 for each hat ad $8 for each scarf. You sell a total of 4 items at the craft fair ad your reveue is $70. How may hats did you sell? (p. 60)

20 .3 Aalyze Geometric Sequeces ad Series Before You studied arithmetic sequeces ad series. Now You will study geometric sequeces ad series. Why? So you ca solve problems about sports touramets, as i Ex. 8. Key Vocabulary geometric sequece commo ratio geometric series I a geometric sequece, the ratio of ay term to the previous term is costat. This costat ratio is called the commo ratio ad is deoted by r. E XAMPLE Idetify geometric sequeces Tell whether the sequece is geometric. a. 4, 0, 8, 8, 40,... b. 6,,,,,... Solutio To decide whether a sequece is geometric, fid the ratios of cosecutive terms. a. a 0 a 4 a 3 8 a 0 9 a 4 8 a a 40 a c The ratios are differet, so the sequece is ot geometric. b. a a 6 a 3 a a 4 a 3 a a 4 c Each ratio is, so the sequece is geometric. GUIDED PRACTICE for Example Tell whether the sequece is geometric. Explai why or why ot.. 8, 7, 9, 3,,....,, 6, 4, 0, , 8, 6, 3, 64,... KEY CONCEPT For Your Notebook Rule for a Geometric Sequece Algebra Example The th term of a geometric sequece with first term a ad commo ratio r is give by: a r The th term of a geometric sequece with a first term of 3 ad commo ratio is give by: 3() 80 Chapter Sequeces ad Series

21 E XAMPLE Write a rule for the th term Write a rule for the th term of the sequece. The fid a 7. a. 4, 0, 00, 00,... b., 76, 38, 9,... Solutio AVOID ERRORS I the geeral rule for a geometric sequece, ote that the expoet is, ot. a. The sequece is geometric with first term a 4 ad commo ratio r 0. So, a rule for the th term is: 4 a r Write geeral rule. 4() Substitute 4 for a ad for r. The 7th term is a 7 4() 7 6,00. b. The sequece is geometric with first term a ad commo ratio r 76. So, a rule for the th term is: a r Write geeral rule. The 7th term is a Substitute for a ad for r. E XAMPLE 3 Write a rule give a term ad commo ratio Oe term of a geometric sequece is a 4. The commo ratio is r. a. Write a rule for the th term. b. Graph the sequece. Solutio a. Use the geeral rule to fid the first term. a r Write geeral rule. a 4 a r 4 Substitute 4 for. a () 3 Substitute for a 4 ad for r.. a Solve for a. So, a rule for the th term is: a r Write geeral rule..() Substitute. for a ad for r. b. Create a table of values for the sequece. The graph of the first 6 terms of the sequece is show. Notice that the poits lie o a expoetial curve. This is true for ay geometric sequece with r > at classzoe.com.3 Aalyze Geometric Sequeces ad Series 8

22 E XAMPLE 4 Write a rule give two terms Two terms of a geometric sequece are a 3 48 ad a Fid a rule for the th term. Solutio STEP Write a system of equatios usig a r ad substitutig 3 for (Equatio ) ad the 6 for (Equatio ). a 3 a r 3 48 a r Equatio a 6 a r a r Equatio STEP Solve the system. 48 r a Solve Equatio for a r (r ) Substitute for a i Equatio r 3 Simplify. 4 r Solve for r. 48 a (4) Substitute for r i Equatio. 3 a Solve for a. STEP 3 Fid a rule for. a r Write geeral rule. 3(4) Substitute for a ad r. GUIDED PRACTICE for Examples, 3, ad 4 Write a rule for the th term of the geometric sequece. The fid a ,, 7, 37,.... a 6 96, r 6. a, a 4 3 GEOMETRIC SERIES The expressio formed by addig the terms of a geometric sequece is called a geometric series. The sum of the first terms of a geometric series is deoted by S. You ca develop a rule for S as follows. S a a r a r a r 3... a r rs a r a r a r 3... a r a r S ( r) a a r So, S ( r) a ( r ). If r Þ, you ca divide each side of this equatio by r to obtai the followig rule for S. KEY CONCEPT For Your Notebook The Sum of a Fiite Geometric Series The sum of the first terms of a geometric series with commo ratio r Þ is: S a r r 8 Chapter Sequeces ad Series

23 E XAMPLE Fid the sum of a geometric series Fid the sum of the geometric series a 4(3) 4 r 3 6 Idetify first term. 4(3) i. Idetify commo ratio. S 6 a r6 r Write rule for S Substitute 4 for a ad 3 for r. 86,093,440 Simplify. c The sum of the series is 86,093,440. E XAMPLE 6 Use a geometric sequece ad series i real life MOVIE REVENUE I 990, the total box office reveue at U.S. movie theaters was about $.0 billio. From 990 through 003, the total box office reveue icreased by about.9% per year. a. Write a rule for the total box office reveue (i billios of dollars) i terms of the year. Let represet 990. b. What was the total box office reveue at U.S. movie theaters for the etire period ? Solutio a. Because the total box office reveue icreased by the same percet each year, the total reveues from year to year form a geometric sequece. Use a.0 ad r to write a rule for the sequece..0(.09) Write a rule for. b. There are 4 years i the period , so fid S 4. S 4 a r4 r.0 (.09)4.09 ø 0 c The total movie box office reveue for the period was about $0 billio. GUIDED PRACTICE for Examples ad 6 7. Fid the sum of the geometric series 8 6() i. 8. MOVIE REVENUE Use the rule i part (a) of Example 6 to estimate the total box office reveue at U.S. movie theaters i Aalyze Geometric Sequeces ad Series 83

24 .3 EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS for Exs. 9, 49, ad 9 STANDARDIZED TEST PRACTICE Exs., 7, 4,, ad 9 MULTIPLE REPRESENTATIONS Ex. 6. VOCABULARY Copy ad complete: The costat ratio of cosecutive terms i a geometric sequece is called the?.. WRITING How ca you determie whether a sequece is geometric? EXAMPLE o p. 80 for Exs. 3 4 IDENTIFYING GEOMETRIC SEQUENCES Tell whether the sequece is geometric. Explai why or why ot. 3., 4, 8, 6, 3, , 6, 64, 6, 04,.... 6, 36, 6,, 6, , 3, 4 3, 8 3, 6 3,... 7.,, 3,,, , 3 8, 3 6, 3, 3 64, ,,.,., 0.6, , 6,, 4, 48,.... 4,, 36, 08, 34, , 0.6,.8,.4, 6.,... 3., 0, 0, 40, 80, ,.,., 3, 3.7,... EXAMPLE o p. 8 for Exs. 7 WRITING RULES Write a rule for the th term of the geometric sequece. The fid a 7.., 4, 6, 64, , 8, 4, 6, , 4, 44, 864, , 3, 7, 87,... 9., 3, 9 8, 7 3, , 6,, 4,.... 4,,, 0., , 0.6,.,.4,... 3., 0.8, 0.3, 0.8, , 4.,.,.,...., 4, 39., 09.76, , 80, 70, 40, MULTIPLE CHOICE What is a rule for the th term of the geometric sequece, 0, 80, 30,...? A () B (4) C (4) D () EXAMPLE 3 o p. 8 for Exs WRITING RULES Write a rule for the th term of the geometric sequece. The graph the first six terms of the sequece. 8. a, r 3 9. a, r a 6, r 3. a, r 3. a, r a 4, r a 3 7, r 3. a 8, r a 4 00, r ERROR ANALYSIS Describe ad correct the error i writig the rule for the th term of the geometric sequece for which a 3 ad r. 37. a r 3() 38. ra (3) 84 Chapter Sequeces ad Series

25 EXAMPLE 4 o p. 8 for Exs WRITING RULES Write a rule for the th term of the geometric sequece that has the two give terms. 39. a 3, a a, a 6 4. a 4, a a 3 0, a a 40, a a 4, a a 4 6, a a 3 7 4, a a 4 6, a EXAMPLE o p. 83 for Exs FINDING SUMS Fid the sum of the geometric series () i i (4) i 0. i 3. 7 i 0 0 (4) i i 0 i 4. MULTIPLE CHOICE What is the sum of the geometric series 9 (3) i? A 9,680 B 9,68 C 9,68 D 9,683. OPEN-ENDED MATH Write a geometric series with terms such that the sum of the series is 00. (Hit: Choose a value of r ad the fid a.) 6. CHALLENGE Usig the rule for the sum of a fiite geometric series, write each polyomial as a ratioal expressio. a. x x x 3 x 4 b. 3x 6x 3 x 4x 7 PROBLEM SOLVING EXAMPLE 6 o p. 83 for Exs SKYDIVING I a skydivig formatio with R rigs, each rig after the first has twice as may skydivers as the precedig rig. The formatio for R is show. a. Let be the umber of skydivers i the th rig. Fid a rule for. b. Fid the total umber of skydivers if there are R 4 rigs. Secod rig First rig 8. SOCCER A regioal soccer touramet has 64 participatig teams. I the first roud of the touramet, 3 games are played. I each successive roud, the umber of games played decreases by oe half. a. Fid a rule for the umber of games played i the th roud. For what values of does your rule make sese? b. Fid the total umber of games played i the regioal soccer touramet..3 Aalyze Geometric Sequeces ad Series 8

26 9. SHORT RESPONSE A biary search techique used o a computer ivolves jumpig to the middle of a ordered list of data (such as a alphabetical list of ames) ad decidig whether the item beig searched for is there. If ot, the computer decides whether the item comes before or after the middle. Half of the list is igored o the ext pass, ad the computer jumps to the middle of the remaiig list. This is repeated util the item is foud. a. Fid a rule for the umber of items remaiig after the th pass through a ordered list of 04 items. b. I the worst case, the item to be foud is the oly oe left i the list after passes through the list. What is the worst-case value of for a biary search of a list with 04 items? Explai. 60. FRACTALS The Sierpiski carpet is a fractal created usig squares. The process ivolves removig smaller squares from larger squares. First, divide a large square ito ie cogruet squares. Remove the ceter square. Repeat these steps for each smaller square, as show below. Assume that each side of the iitial square is oe uit log. Stage Stage Stage 3 a. Let be the umber of squares removed at the th stage. Fid a rule for. The fid the total umber of squares removed through stage 8. b. Let b be the remaiig area of the origial square after the th stage. Fid a rule for b. The fid the remaiig area of the origial square after stage. 6. MULTIPLE REPRESENTATIONS Two compaies, compay A ad compay B, offer the same startig salary of $0,000 per year. Compay A gives a raise of $000 each year. Compay B gives a raise of 4% each year. a. Writig Rules Write rules givig the salaries ad b i the th year at compaies A ad B, respectively. Tell whether the sequece represeted by each rule is arithmetic, geometric, or either. b. Drawig Graphs Graph each sequece i the same coordiate plae. c. Fidig Sums For each compay, fid the sum of wages eared durig the first 0 years of employmet. d. Usig Techology Use a graphig calculator or spreadsheet to fid after how may years the total amout eared at compay B is greater tha the total amout eared at compay A. WORKED-OUT SOLUTIONS 86 Chapter Sequeces o p. WS ad Series STANDARDIZED TEST PRACTICE MULTIPLE REPRESENTATIONS

27 6. CHALLENGE O Jauary of each year, you deposit $000 i a idividual retiremet accout (IRA) that pays % aual iterest. You make a total of 30 deposits. How much moey do you have i your IRA immediately after you make your last deposit? MIXED REVIEW Graph the umbers o umber lie. (p. ) 63., 9, 3,, Ï 6, 3, 4 7,, 6. 4,.7,.8, 7, Ï 8 Solve the equatio. Check your solutio. (p. 89) x x x x x x 6 x x x 7. x 6 x x 8 PREVIEW Prepare for Lesso.4 i Exs Fid the sum of the series. (p. 794) i i 7 (i ) 73. (3i 4) 76. i 3 i 6i 74. (i ) i 4 ( i) 4i QUIZ for Lessos..3 Write the ext term i the sequece. The write a rule for the th term. (p. 794)., 3,, 7,...., 0,, 0, , 30, 3 40, 4 0, , 6, 64, 6,...., 6,, 0, , 36, 8, 44,... Fid the sum of the series. (p. 794) 7. 4 i 3 8. k (k 3) 9. 6 Write a rule for the th term of the arithmetic or geometric sequece. Fid a, the fid the sum of the first terms of the sequece. 0., 7, 3, 9,... (p. 80).,, 7,,... (p. 80).,,, 4, 7,... (p. 80) 3., 8, 3, 8,... (p. 80) 4., 4 3, 8 9, 6,... (p. 80). 3,, 7, 37,... (p. 80) 7 6. COLLEGE TUITION I 99, the average tuitio at a public college i the Uited States was $07. From 99 through 00, the average tuitio at public colleges icreased by about 6% per year. Write a rule for the average tuitio i terms of the year. Let represet 99. What was the average tuitio at a public college i 00? (p. 80) EXTRA PRACTICE for Lesso.3, p. 0.3 Aalyze ONLINE Geometric QUIZ Sequeces at classzoe.com ad Series 87

28 MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos..3. MULTI-STEP PROBLEM You accept a job as a evirometal egieer that pays a salary of $4,000 i the first year. After the first year, your salary icreases by 3.% per year. a. Write a rule givig your salary for your th year of employmet. b. What will your salary be durig your th year of employmet? c. What is the total amout you will ear if you work for the compay for 30 years? 4. OPEN-ENDED Write a arithmetic series with eight terms such that the sum of the series is 70.. GRIDDED ANSWER Pieces of chalk are stacked i a pile. Part of the pile is show below.. EXTENDED RESPONSE A target has rigs that are each foot wide. The 3 iermost rigs of the target ft The bottom row has pieces of chalk ad the top row has 6 pieces of chalk. Each row has oe less piece of chalk tha the row below it. How may pieces of chalk are i the pile? a. Write a rule for the area of the th rig. b. Use summatio otatio to write a series that gives the total area of a target with rigs. c. Evaluate your expressio from part (b) whe,, 4, ad 8. What effect does doublig the umber of rigs have o the area of the target? 3. SHORT RESPONSE Rectagular tables are placed together alog their short edges, as show i the diagram. Write a rule for the umber of people that ca be seated aroud tables. Compare the umber of people that ca be seated aroud tables i this arragemet with the umber that ca be seated aroud tables if the same tables are placed together alog their log edges. 6. SHORT RESPONSE A builder is costructig a staircase for a deck. At the foot of the staircase, there is a cocrete slab that is iches tall. Each stair is 7 iches tall. Write a rule for the height of the top of the th stair. Fid the height of the top of the 0th stair. Explai how you could modify the rule so that it gives the height of the bottom of the th stair. 7. EXTENDED RESPONSE A scietist is studyig the radioactive decay of Platium-97. The scietist starts with a 66 gram sample of Platium-97 ad measures the amout remaiig every two hours. The amouts (i grams) recorded are 66, 33, 6., 8.,.... a. Is this sequece arithmetic, geometric, or either? Explai how you kow. b. Write a rule for the th term of the sequece. c. Graph the sequece. Describe the curve o which the poits lie. d. After how may hours will the scietist first measure a amout of Platium-97 that is less tha gram? 8. OPEN-ENDED Write a arithmetic series ad a geometric series such that both series have five terms ad the same sum. 88 Chapter Sequeces ad Series

29 Use before Lesso.4.4 Ivestigatig a Ifiite Geometric Series MATERIALS scissors paper QUESTION What is the sum of a ifiite geometric series? You ca illustrate a ifiite geometric series by cuttig a piece of paper ito smaller ad smaller pieces. E XPLORE Model a ifiite geometric series Start with a rectagular piece of paper. Defie its area to be square uit. STEP Cut paper i half STEP Cut paper agai STEP 3 Repeat steps Fold the paper i half ad cut alog the fold. Place oe half o a desktop ad hold the remaiig half. Fold the piece of paper you are holdig i half ad cut alog the fold. Place oe half o the desktop ad hold the remaiig half. Repeat Steps ad util you fid it too difficult to fold ad cut the piece of paper you are holdig. STEP 4 Fid areas The first piece of paper o the desktop has a area of square uit. The secod piece has a area of square uit. Write the areas 4 of the ext three pieces of paper. Explai why these areas form a geometric sequece. STEP Make a table Copy ad complete the table by recordig the umber of pieces of paper o the desktop ad the combied area of the pieces at each step. Number of pieces Combied area 4???... DRAW CONCLUSIONS Use your observatios to complete these exercises. Based o your table, what umber does the combied area of the pieces of paper appear to be approachig?. Usig the formula for the sum of a fiite geometric series, write ad simplify a rule for the combied area A of the pieces of paper after cuts. What happes to A as? Justify your aswer mathematically.

30 .4 Fid Sums of Ifiite Geometric Series Before You foud the sums of fiite geometric series. Now You will fid the sums of ifiite geometric series. Why? So you ca aalyze a fractal, as i Ex. 4. Key Vocabulary partial sum The sum S of the first terms of a ifiite series is called a partial sum. The partial sums of a ifiite geometric series may approach a limitig value. E XAMPLE Fid partial sums Cosider the ifiite geometric series Fid ad graph the partial sums S for,, 3, 4, ad. The describe what happes to S as icreases. Solutio S 0. S S < 0.88 S ø 0.94 S ø S From the graph, S appears to approach as icreases. at classzoe.com SUMS OF INFINITE SERIES I Example, you ca uderstad why S approaches as icreases by cosiderig the rule for S : S a r r As icreases, approaches 0, so S approaches. Therefore, is defied to be the sum of the ifiite geometric series i Example. More geerally, as icreases for ay ifiite geometric series with commo ratio r betwee ad, the value of S a r r ø a 0 r a r. 80 Chapter Sequeces ad Series

31 KEY CONCEPT For Your Notebook The Sum of a Ifiite Geometric Series The sum of a ifiite geometric series with first term a ad commo ratio r is give by S a r provided r <. If r, the series has o sum. E XAMPLE Fid sums of ifiite geometric series Fid the sum of the ifiite geometric series. a. Solutio (0.8) i b a. For this series, a ad r 0.8. b. For this series, a ad r 3 4. a S r 0.8 S a r AVOID ERRORS If you substitute for a ad 3 for r i the formula S a r, you get a aswer of S 4 for the sum. However, this aswer is ot correct because the sum formula does ot apply whe r. E XAMPLE 3 Solutio Stadardized Test Practice What is the sum of the ifiite geometric series ? A 4 B You kow that a ad a 3. So, r 3 3. Because 3, the sum does ot exist. c The correct aswer is D. A B C D 4 3 C 3 D Does ot exist GUIDED PRACTICE for Examples,, ad 3. Cosider the series Fid ad graph the partial sums S for,, 3, 4, ad. The describe what happes to S as icreases. Fid the sum of the ifiite geometric series, if it exists Fid Sums of Ifiite Geometric Series 8

32 E XAMPLE 4 Use a ifiite series as a model PENDULUMS A pedulum that is released to swig freely travels 8 iches o the first swig. O each successive swig, the pedulum travels 80% of the distace of the previous swig. What is the total distace the pedulum swigs? Solutio The total distace traveled by the pedulum is: d 8 8(0.8) 8(0.8) 8(0.8) 3... a r Simplify. Write formula for sum. Substitute 8 for a ad 0.8 for r. c The pedulum travels a total distace of 90 iches, or 7. feet. E XAMPLE Write a repeatig decimal as a fractio Write as a fractio i lowest terms (0.0) 4(0.0) 4(0.0) 3... a r 4(0.0) Write formula for sum. Substitute 4(0.0) for a ad 0.0 for r. Simplify. Write as a quotiet of itegers. Reduce fractio to lowest terms. c The repeatig decimal is 8 as a fractio. 33 GUIDED PRACTICE for Examples 4 ad. WHAT IF? I Example 4, suppose the pedulum travels 0 iches o its first swig. What is the total distace the pedulum swigs? Write the repeatig decimal as a fractio i lowest terms Chapter Sequeces ad Series

33 .4 EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS for Exs. 3, 7, ad 39 STANDARDIZED TEST PRACTICE Exs., 3, 34, 39, 40, ad 4 EXAMPLE o p. 80 for Exs VOCABULARY Copy ad complete: The sum S of the first terms of a ifiite series is called a()?.. WRITING Explai how to tell whether the series a r i has a sum. PARTIAL SUMS For the give series, fid ad graph the partial sums S for,, 3, 4, ad. Describe what happes to S as icreases EXAMPLES ad 3 o p. 8 for Exs. 7 3 FINDING SUMS Fid the sum of the ifiite geometric series, if it exists i k 8.. 9(4) k k k i 7. 3 i k 9 k 4. i i 8. k k (3) 9. ERROR ANALYSIS Describe ad correct the error i fidig the sum of the ifiite geometric series 7. For this series, a ad r 7. a S r 7 FINDING SUMS Fid the sum of the ifiite geometric series, if it exists EXAMPLE o p. 8 for Exs. 4 3 REWRITING DECIMALS Write the repeatig decimal as a fractio i lowest terms MULTIPLE CHOICE Which fractio is equal to the repeatig decimal ? A B C 00 D REASONING Show that is equal to..4 Fid Sums of Ifiite Geometric Series 83

34 34. OPEN-ENDED MATH Fid two ifiite geometric series whose sums are each. CHALLENGE Specify the values of x for which the give ifiite geometric series has a sum. The fid the sum i terms of x. 3. 4x 6x 64x x 3 8 x 3 3 x3... PROBLEM SOLVING EXAMPLE 4 o p. 8 for Exs TIRE SWING A perso is give oe push o a tire swig ad the allowed to swig freely. O the first swig, the perso travels a distace of 4 feet. O each successive swig, the perso travels 80% of the distace of the previous swig. What is the total distace the perso swigs? 38. BUSINESS A compay had a profit of $30,000 i its first year. Sice the, the compay s profit has decreased by % per year. If this tred cotiues, what is a upper limit o the total profit the compay ca make over the course of its lifetime? Justify your aswer usig a ifiite geometric series. 39. MULTIPLE CHOICE I 994, the umber of cassette tapes shipped i the Uited States was 34 millio. I each successive year, the umber decreased by about.7%. What is the total umber of cassettes that will ship i 994 ad after if this tred cotiues? A 40 millio B 440 millio C 6 millio D.9 billio 40. SHORT RESPONSE Ca the Greek hero Achilles, ruig at 0 feet per secod, ever catch up to a tortoise that rus 0 feet per secod if the tortoise has a 0 foot head start? The Greek mathematicia Zeo said o. He reasoed as follows: Whe Achilles rus 0 feet, The, whe Achilles gets to Achilles will keep halvig the tortoise will be i ew that spot, the tortoise will the distace but will ever spot, 0 feet away. be feet away. catch up to the tortoise. I actuality, lookig at the race as Zeo did, you ca see that both the distaces ad the times Achilles required to traverse them form ifiite geometric series. Usig the table, show that both series have fiite sums. Does Achilles catch up to the tortoise? Explai. Distace (ft) Time (sec) WORKED-OUT SOLUTIONS 84 Chapter Sequeces o p. WS ad Series STANDARDIZED TEST PRACTICE

35 4. EXTENDED RESPONSE A studet drops a rubber ball from a height of 8 feet. Each time the ball hits the groud, it bouces to 7% of its previous height. a. How far does the ball travel betwee the first ad secod bouces? betwee the secod ad third bouces? b. Write a ifiite series to model the total distace traveled by the ball, excludig the distace traveled before the first bouce. c. Fid the total distace traveled by the ball, icludig the distace traveled before the first bouce. 3 4 Bouce umber d. Show that if the ball is dropped from a height of h feet, the the total distace traveled by the ball (icludig the distace traveled before the first bouce) is 7h feet. 4. CHALLENGE The Sierpiski triagle is a fractal created usig equilateral triagles. The process ivolves removig smaller triagles from larger triagles by joiig the midpoits of the sides of the larger triagles as show below. Assume that the iitial triagle has a area of square uit. 8 ft 6 ft 6 ft 4. ft 4. ft 3.37 ft 3.37 ft.3 ft.3 ft a. Let be the total area of all the triagles that are removed at stage. Write a rule for. Stage Stage Stage 3 b. Fid. What does your aswer mea i the cotext of this problem? MIXED REVIEW Fid the idicated probability. (p. 707) 43. P(A) 3% 44. P(A)? 4. P(A) 0. P(B)? P(B) 0.7 P(B) 0.4 P(A or B) 8% P(A or B) 0.6 P(A or B) 0. P(A ad B) % P(A ad B) 0.03 P(A ad B)? PREVIEW Prepare for Lesso. i Exs Write a rule for the th term of the arithmetic sequece with commo differece d or the geometric sequece with commo ratio r. 46. d, a (p. 80) 47. d 8, a 7 (p. 80) 48. d 8, a 8 7 (p. 80) 49. d 7, a 7 8 (p. 80) 0. d 6., a 9 (p. 80). d., a 9 4 (p. 80). r., a 3 (p. 80) 3. r 3, a 8 (p. 80) 4. r 0., a 40. (p. 80). r 6, a 4 4 (p. 80) 6. r 0.7, a 6 30 (p. 80) 7. r 4, a 4 8 (p. 80) EXTRA PRACTICE for Lesso.4, p. 0.4 ONLINE Fid Sums QUIZ of Ifiite at classzoe.com Geometric Series 8

36 Ivestigatig g Algebra ACTIVITY. Explorig Recursive Rules MATERIALS computer with spreadsheet program Use before Lesso. classzoe.com Keystrokes QUESTION How ca you evaluate a recursive rule for a sequece? A recursive rule for a sequece gives the begiig term or terms of the sequece ad the a equatio relatig the th term to oe or more precedig terms. For example, the rule a 4, 7 defies a sequece recursively. EXPLORE Fid terms of a sequece give by a recursive rule Fid the first eight terms of the sequece defied by a 4, 7. What type of sequece does this rule represet? STEP Eter first term Eter the value of a ito cell A. STEP Eter recursive equatio Eter the formula A7 ito cell A. STEP 3 Fill cells Use the fill dow commad to copy the recursive equatio ito the rest of colum A. A A 4 B C A A7 A B 4 C A A77 A B C STEP 4 Idetify terms ad type of sequece The first eight terms of the sequece are 4,, 8,, 3, 39, 46, ad 3. This sequece is a arithmetic sequece because the differece of cosecutive terms is always 7. DRAW CONCLUSIONS Use your observatios to complete these exercises. Fid the first eight terms of the sequece defied by a 4, 7. What type of sequece does this rule represet?. Write a recursive rule for the sequece,, 7, 3,,, Write a recursive rule for the sequece 8, 7, 9, 3,, 3, What equatio relates the th term to the precedig term for a arithmetic sequece with commo differece d? for a geometric sequece with commo ratio r? 86 Chapter Sequeces ad Series

37 . Use Recursive Rules with Sequeces ad Fuctios Before You used explicit rules for sequeces. Now You will use recursive rules for sequeces. Why? So you ca model evaporatio from a pool, as i Ex. 44. Key Vocabulary explicit rule recursive rule iteratio So far i this chapter you have worked with explicit rules for the th term of a sequece, such as 3 ad 3(). A explicit rule gives as a fuctio of the term s positio umber i the sequece. I this lesso you will lear aother way to defie a sequece by a recursive rule. A recursive rule gives the begiig term or terms of a sequece ad the a recursive equatio that tells how is related to oe or more precedig terms. E XAMPLE Evaluate recursive rules Write the first six terms of the sequece. a. a 0, 4 b. a, 3 Solutio a. a 0 b. a a a a 3a 3() 3 a a a 3 3a 3(3) 9 a 3 a a 4 3a 3 3(9) 7 a 4 a a 3a 4 3(7) 8 a a a 6 3a 3(8) 43 ARITHMETIC AND GEOMETRIC SEQUENCES I part (a) of Example, observe that the differeces of cosecutive terms of the sequece are costat, so the sequece is arithmetic. I part (b), the ratios of cosecutive terms are costat, so the sequece is geometric. I geeral, rules for arithmetic ad geometric sequeces ca be writte recursively as follows. KEY CONCEPT For Your Notebook Recursive Equatios for Arithmetic ad Geometric Sequeces Arithmetic Sequece d where d is the commo differece Geometric Sequece r p where r is the commo ratio. Use Recursive Rules with Sequeces ad Fuctios 87

38 E XAMPLE Write recursive rules Write a recursive rule for the sequece. a. 3, 3, 3, 33, 43,... b. 6, 40, 00, 0, 6,... Solutio AVOID ERRORS A recursive equatio for a sequece does ot iclude the iitial term. To write a recursive rule for a sequece, the iitial term must be icluded. a. The sequece is arithmetic with first term a 3 ad commo differece d d Geeral recursive equatio for 0 Substitute 0 for d. c So, a recursive rule for the sequece is a 3, 0. b. The sequece is geometric with first term a 6 ad commo ratio r r p Geeral recursive equatio for. Substitute. for r. c So, a recursive rule for the sequece is a 6,.. GUIDED PRACTICE for Examples ad Write the first five terms of the sequece.. a 3, 7. a 0 6, a 0, 4. a 4, Write a recursive rule for the sequece.., 4, 98, 686, 480, , 3, 7,,,... 7.,, 33, 44,, , 08, 36,, 4,... RECURSIVE RULES FOR SPECIAL SEQUENCES For some sequeces, it is difficult to write a explicit rule but relatively easy to write a recursive rule. E XAMPLE 3 Write recursive rules for special sequeces Write a recursive rule for the sequece. a.,,, 3,,... b.,,, 6, 4,... NAME SEQUENCES The sequece i part (a) of Example 3 is called the Fiboacci sequece. The sequece i part (b) of Example 3 lists the factorial umbers you studied i Chapter 0. Solutio a. Begiig with the third term i the sequece, each term is the sum of the two previous terms. c So, a recursive rule is a, a,. b. Deote the first term by a 0. The ote that a p a 0, a p a, a p a, ad so o. c So, a recursive rule is a 0, p. 88 Chapter Sequeces ad Series

39 E XAMPLE 4 Solve a multi-step problem MUSIC SERVICE A olie music service iitially has 0,000 aual members. Each year it loses 0% of its curret members ad adds 000 ew members. Write a recursive rule for the umber of members at the start of the th year. Fid the umber of members at the start of the th year. Describe what happes to the umber of members over time. ANOTHER WAY For alterative methods for solvig the problem i Example 4, tur to page 834 for the Problem Solvig Workshop. Solutio STEP Write a recursive rule. Because the umber of members declies 0% each year, 80% of the members are retaied from oe year to the ext. Also, 000 ew members are added each year. Members at start of year 0.8 p Members at start of year ( ) New members added 0.8 p 000 STEP c A recursive rule is a 0,000, Fid the umber of members at the start of the th year. Eter 0,000 (the value of a ) ito a graphig calculator. The eter the rule As 000 to fid a. Press three more times to fid a. c There are about 3,40 members at the start of the th year *As STEP 3 Describe what happes to the umber of members over time. Cotiue pressig o the calculator. As show at the right, after may years the umber of members approaches,000. c The umber of members stabilizes at about,000 members GUIDED PRACTICE for Examples 3 ad 4 9. Write a recursive rule for the sequece,,, 4, 8, 3, WHAT IF? I Example 4, suppose 70% of the members are retaied each year. What happes to the umber of members over time?. Use Recursive Rules with Sequeces ad Fuctios 89

40 ITERATING FUNCTIONS Iteratio ivolves the repeated compositio of a fuctio f with itself. The result of oe iteratio is f(f(x)). The result of two iteratios is f(f(f(x))). You ca use iteratio to geerate a sequece recursively. Begi with a iitial value x 0, ad let x f(x 0 ), x f(x ) f(f(x 0 )), ad so o. READING A iterate is umber that is the result of iteratig a fuctio. E XAMPLE Iterate a fuctio Fid the first three iterates x, x, ad x 3 of the fuctio f(x) 3x for a iitial value of x 0. Solutio x f(x 0 ) x f(x ) x 3 f(x ) f() 3() f() 3() 6 f(6) 3(6) 47 c The first three iterates are, 6, ad 47. GUIDED PRACTICE for Example Fid the first three iterates of the fuctio for the give iitial value.. f(x) 4x 3, x 0. f(x) x, x 0. EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS p. WS for Exs., 7, ad 4 STANDARDIZED TEST PRACTICE Exs.,, 33, 40, 4, ad 47. VOCABULARY Copy ad complete: The repeated compositio of a fuctio with itself is called?.. WRITING Explai the differece betwee a explicit rule for a sequece ad a recursive rule for a sequece. EXAMPLE o p. 87 for Exs. 3 WRITING TERMS Write the first five terms of the sequece. 3. a 4. a 0 4. a 3 6. a a 8. a 0 4 ( ) ( ) 0 9. a 0. a 0, a 4. a, a 3 3 p. MULTIPLE CHOICE What are the first four terms of the sequece for which a, a 4, ad p? A, 4, 4, 6 B, 4, 6, 64 C, 4, 8, 6 D, 4, 4, Chapter Sequeces ad Series

41 EXAMPLES ad 3 o p. 88 for Exs. 3 3 WRITING RULES Write a recursive rule for the sequece. The sequece may be arithmetic, geometric, or either. 3., 4, 7, 0, 7, ,, 48, 9, 768,.... 4,, 36, 08, 34,... 6., 8,,, 9, ,, 4, 6,,... 8., 4,, 9, 4, , 43, 3,, 0, ,,, 7,,.... 6, 9, 7,,,... ERROR ANALYSIS Describe ad correct the error i writig a recursive rule for the sequece,, 3,, 4,..... Begiig with the third term i the sequece, each term equals. So a recursive rule is give by: 3. Begiig with the secod term i the sequece, each term is 3. So a recursive rule is give by: a, 3 EXAMPLE o p. 830 for Exs ITERATING FUNCTIONS Fid the first three iterates of the fuctio for the give iitial value. 4. f(x) 3x, x 0. f(x) x 6, x 0 6. g(x) 4x 7, x 0 7. f(x) x 3, x 0 8. f(x) 3 x, x h(x) x 4, x f(x) x, x 0 3. f(x) x x, x 0 3. g(x) 3x x, x MULTIPLE CHOICE What are the first three iterates x, x, ad x 3 of the fuctio f(x) x 3 for a iitial value of x 0? A,, 3 B,, 7 C,, 7 D,, 3 WRITING RULES Write a recursive rule for the sequece , 8, 7, 8, 370,... 3.,,, 6, 7, , Ï 3,, Ï 3, 4, ,,, 6, 9, , 4,,,, ,,, 3,, OPEN-ENDED MATH Give a example of a sequece i which each term after the third term is a fuctio of the three terms precedig it. Write a recursive rule for the sequece ad fid its first eight terms. 4. REASONING Explai why there are ot a fuctio f ad a iitial value x 0 such that the fuctio s first three iterates are x, x, ad x CHALLENGE You ca defie a sequece usig a piecewise rule. The followig is a example of a piecewise-defied sequece. a,, if a is eve 3 3, if is odd a. Write the first te terms of the sequece. b. Choose three differet positive iteger values for a (other tha a ). For each value of a, fid the first te terms of the sequece. What coclusios ca you make about the behavior of this sequece of itegers?. Use Recursive Rules with Sequeces ad Fuctios 83

42 PROBLEM SOLVING EXAMPLE 4 o p. 89 for Exs FISH POPULATION A lake iitially cotais 000 fish. Each year the populatio declies 0% due to fishig ad other causes, ad the lake is restocked with 00 fish. a. Write a recursive rule for the umber of fish at the begiig of the th year. How may fish are there at the begiig of the th year? b. What happes to the populatio of fish i the lake over time? 44. POOL CARE You are addig chlorie to a swimmig pool. You add 34 ouces of chlorie the first week ad 6 ouces every week thereafter. Each week 40% of the chlorie i the pool evaporates. Write a recursive rule for the amout of chlorie i the pool each week. What happes to the amout of chlorie i the pool over time? First week 34 oz of chlorie are added Each successive week 6 oz of chlorie are added 40% of chlorie has evaporated 4. SHORT RESPONSE Gladys owes $000 to a credit card compay that charges iterest at a rate of.4% per moth. At the ed of each moth she makes a paymet of $00. Write a recursive rule for the balace of the accout at the begiig of the th moth. How log will it take to pay off the accout? Explai your reasoig. 46. FIBONACCI SEQUENCE The Fiboacci sequece, which is defied recursively i Example 3 o page 88, occurs may places i ature. This sequece ca also be defied explicitly as follows: f Ï Ï Ï Ï, Use the explicit rule to fid the first five terms of the Fiboacci sequece. at classzoe.com 47. EXTENDED RESPONSE A perso repeatedly takes 0 milligrams of a prescribed drug every 4 hours. Thirty percet of the drug is removed from the bloodstream every 4 hours. a. Write a recursive rule for the amout of the drug i the bloodstream after doses. b. The value that a drug level i a perso s body approaches after a exteded period of time is called the maiteace level. What is the maiteace level of this drug, give a dosage of 0 milligrams? c. How does doublig the dosage affect the maiteace level of the drug? Justify your aswer mathematically. WORKED-OUT SOLUTIONS 83 Chapter Sequeces o p. WS ad Series STANDARDIZED TEST PRACTICE

43 48. CHALLENGE You are savig moey for retiremet. You pla to withdraw $30,000 at the begiig of each year for 0 years after you retire. Based o the type of ivestmet you are makig, you ca expect to ear a aual retur of 8% o your savigs after you retire. a. Let be your balace years after retirig. Write a recursive equatio that shows how is related to. b. Solve the equatio from part (a) for. Fid a 0, the miimum amout of moey you should have i your accout whe you retire. (Hit: Let a 0 0.) MIXED REVIEW PREVIEW Prepare for Lesso 3. i Exs. 49. Fid the value of x. (p. 99) 49. x 3 cm 3 cm 0. ft x. 9 ft m x m Evaluate the expressio without usig a calculator. (p. 44). 6 3/ 3. (43) / / QUIZ for Lessos.4. Fid the sum of the ifiite geometric series, if it exists. (p. 80) Write the repeatig decimal as a fractio i lowest terms. (p. 80) Write the first five terms of the sequece. (p. 87) 7. a 8. a a, a 4 4 ( ) Write a recursive rule for the sequece. The sequece may be arithmetic, geometric, or either. (p. 87) 0., 7 4, 7,,,...., 6,, 7, 864,.... 8, 4, 7, 6, 648,... 4 Fid the first three iterates of the fuctio for the give iitial value. (p. 87) 3. f(x) 3x, x 0 4. g(x) 4x, x 0. f(x) x 3, x 0 6. f(x) x 7, x h(x) x 6, x 0 8. f(x) 3x, x PENDULUMS A pedulum that is released to swig freely travels iches o the first swig. O each successive swig, the pedulum travels 8% as far as the previous swig. What is the total distace the pedulum swigs? (p. 80) EXTRA PRACTICE for Lesso.,. p. 0 Use Recursive ONLINE Rules with QUIZ Sequeces at classzoe.com ad Fuctios 833

44 LESSON. Usig ALTERNATIVE METHODS Aother Way to Solve Example 4, page 89 MULTIPLE REPRESENTATIONS I Example 4 o page 89, you foud the umber that a real-life sequece approaches over time by usig a calculator to evaluate the rule for the sequece. You ca also solve this problem usig a graph or a algebraic method. P ROBLEM MUSIC SERVICE A olie music service iitially has 0,000 aual members. Each year the music service loses 0% of its curret members ad adds 000 ew members. What happes to the umber of members over time? M ETHOD Usig a Graph A recursive rule for the umber of members at the begiig of the th year is a 0,000, Oe alterative method for fidig the umber this sequece approaches is to graph the sequece o a graphig calculator. STEP Set the calculator to sequece mode ad dot mode. Normal Sci Eg Float Radia Degree Fuc Par Pol Seq Coected Dot Sequetial Simul Real a+bi re^ui Full Horiz G-T STEP Press ad eter the equatios Mi, u() 0.8u( ) 000, ad u(mi) 0,000. Press ad eter the followig parameters: Mi Xmi 0 Ymi,000 Max 00 Xmax 00 Ymax 3,000 PlotStart Xscl 0 Yscl 000 PlotStep Mi= u()=.8u(-)+000 u(mi)={0000 v()= v(mi)= w()= w(mi)= STEP 3 Graph the sequece. Use the trace feature to fid the value that the sequece approaches as becomes large. From the graph, you ca see that the sequece approaches,000. =74 X=74 Y=000 c Over time, the umber of members of the music service approaches, Chapter Sequeces ad Series

45 M ETHOD Usig Algebra Aother approach is to use a algebraic method to determie what happes to the umber of members over time. STEP Write the recursive rule. a 0,000, STEP Assume that the sequece has a limit L, which is the value that the sequece approaches as becomes large. STEP 3 Cosider what happes to the equatio as becomes large. The value of (the left-had side) approaches L while the value of (the right-had side) approaches 0.8L 000. So, you ca coclude that L 0.8L 000. STEP 4 Solve the equatio L 0.8L 000 for L. L 0.8L 000 Write equatio. 0.L 000 Subtract 0.8L from each side. L,000 Divide each side by 0.. c The sequece approaches the limit L,000 as becomes large. So, over time the umber of members of the music service approaches,000. P RACTICE Describe what happes to the terms of the sequece as becomes large.. a 3000, a 700, WHAT IF? Suppose the olie music service i the problem o page 834 loses 8% of its curret members ad adds 00 ew members each year. Use the graphig method ad the algebraic method to determie what happes to the umber of members over time. 4. TOWN LIBRARY A tow library iitially has 4,000 books i its collectio. Each year % of the books are lost or discarded. The library ca afford to purchase 0 ew books each year. Write a recursive rule for the umber of books i the library at the begiig of the th year. Use the graphig method ad the algebraic method to determie what happes to the umber of books i the library over time.. ERROR ANALYSIS A studet attempted to solve the problem i Exercise 4 as show below. Describe ad correct the error i the studet s work. a 4,000, Let L be the limit of the sequece. The: L 0.0L L 0 Lø 73 So, over time the umber of books i the library approaches about REASONING Give a example of a real-life situatio which you ca represet with a recursive rule that does ot approach a limit. Write a recursive rule that represets the situatio. Usig Alterative Methods 83

46 Extesio Use after Lesso. Prove Statemets Usig Mathematical Iductio GOAL Use mathematical iductio to prove statemets about all positive itegers. I Lesso., you saw the rule for the sum of the first positive itegers:... ( ) You ca use mathematical iductio to prove statemets about positive itegers. KEY CONCEPT For Your Notebook Mathematical Iductio To show that a statemet is true for all positive itegers, perform these steps. Basis Step: Show that the statemet is true for. Iductive Step: Assume that the statemet is true for k where k is ay positive iteger. Show that this implies the statemet is true for k. E XAMPLE Use mathematical iductio UNDERSTAND INDUCTION If you kow from the basis step that a statemet is true for, the the iductive step implies that it is true for, ad therefore for 3, ad so o for all positive itegers. Use mathematical iductio to prove that... ( ). Solutio Basis Step: Check that the formula works for. 0 ( ) Iductive Step: Assume that... k(k ) k. Show that... (k )[(k ) ] k (k ).... k k(k ) Assume true for k.... k(k ) k (k ) (k ) Add k to each side. k(k ) (k ) (k )(k ) (k )[(k ) ] Add. Factor out k. Rewrite k as (k ). Therefore,... ( ) for all positive itegers. 836 Chapter Sequeces ad Series

47 E XAMPLE Use mathematical iductio Let with a. Use mathematical iductio to prove that a explicit rule for the th term is. Solutio Basis Step: Check that the formula works for. a 0 Iductive Step: Assume that a k k. Show that a k k. a k a k Defiitio of for k k Substitute for a k. k Multiply. k 4 k Add. Simplify. Therefore, a explicit rule for the th term is for all positive itegers. PRACTICE EXAMPLES ad o pp for Exs. 8 Use mathematical iductio to prove the statemet.. 3. (i ). i 4. i ( )( ) 6 a r i a r r. i(i ) 6. (i) ( )( ) 3 7. GEOMETRY The umbers, 6,, 8,... are called hexagoal umbers because they represet the umbers of dots used to make hexagos, as show below. Prove that the th hexagoal umber H is give by H ( ). 8. REASONING Let f, f,..., f,... be the Fiboacci sequece. Prove that f f... f f for all positive itegers. Extesio: Prove Statemets Usig Mathematical Iductio 837

48 MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos.4.. MULTI-STEP PROBLEM A ball is dropped from a height of feet. Each time the ball hits the groud, it bouces to 70% of its previous height.. SHORT RESPONSE Why does the sum of a ifiite geometric series ot exist if r where r is the commo ratio? 6. SHORT RESPONSE The legth l of the first loop of a sprig is 6 iches. The legth l of the secod loop is 0.9 times the legth of the first loop. The legth l 3 of the third loop is 0.9 times the legth of the secod loop, ad so o. If the sprig could have ifiitely may loops, would its legth be fiite or ifiite? Explai. If its legth is fiite, fid the legth. a. Write a ifiite series to model the total distace traveled by the ball, excludig the distace traveled before the first bouce. b. Fid the total distace traveled by the ball, icludig the distace traveled before the first bouce.. MULTI-STEP PROBLEM A fractal tree starts with a sigle brach (the truk). At each stage, the ew braches from the previous stage each grow two more braches as show. Stage Stage Stage 3 a. List the umber of ew braches i each of the first six stages. b. Is the sequece of umbers from part (a) arithmetic, geometric, or either? c. Write a explicit rule ad a recursive rule for the sequece from part (a). 3. GRIDDED ANSWER What is the sum of the first three iterates of the fuctio f(x) x 8 whe the iitial value is x 0? 4. OPEN-ENDED Give a example of a explicit rule for a sequece ad a recursive rule for the same sequece EXTENDED RESPONSE You take out a five year loa of $0,000 to buy a car. The loa has a aual iterest rate of 6.% compouded mothly. Each moth you make a mothly paymet of $96 (except the last moth whe you make a paymet of oly $6). a. Fid the mothly iterest rate. The write a recursive rule for the amout of moey you owe after moths. b. How much moey do you owe after moths? c. Suppose you had decided to pay a additioal $0 with each mothly paymet. Use a graphig calculator to fid the umber of moths you would have eeded to repay the loa. d. I your opiio, is it beeficial to pay the additioal $0 with each paymet? Explai your reasoig. 8. GRIDDED ANSWER A tree farm iitially has 8000 trees. Each year 0% of the trees are harvested ad 00 seedligs are plated. What umber of trees evetually exists o the farm after a exteded period of time? 9. OPEN-ENDED Write a ifiite geometric series that has a sum of Chapter Sequeces ad Series

49 Big Idea CHAPTER SUMMARY BIG IDEAS Aalyze Sequeces For Your Notebook The iformatio below highlights the similarities ad differeces betwee arithmetic ad geometric sequeces. Arithmetic Sequece a ( )d First term: a Geometric Sequece a r First term: a Commo differece: d Commo ratio: r Graph is liear. Graph is expoetial. Big Idea Fid Sums of Series The most commo formulas for sums of series are show below. Arithmetic Series Geometric Series Ifiite Geometric Series Sum of the first terms: S a Example: S Sum of the first terms: S a r r, r Þ Example: S Sum of the series: a S r, r < Example: S Other commo sum formulas: i ( ) i ( )( ) 6 Big Idea 3 Use Recursive Rules The table shows explicit ad recursive rules for arithmetic ad geometric sequeces. Arithmetic Sequece Example: 3,, 7, 9,,... Explicit Rule a ( )d Recursive Rule d a 3, Geometric Sequece Example: 8, 4,,, 0.,... a r 8(0.) r p a 8, 0. Chapter Summary 839

50 CHAPTER REVIEW classzoe.com REVIEW KEY VOCABULARY Multi-Laguage Glossary Vocabulary practice sequece, p. 794 terms of a sequece, p. 794 series, p. 796 summatio otatio, p. 796 sigmotatio, p. 796 arithmetic sequece, p. 80 commo differece, p. 80 arithmetic series, p. 804 geometric sequece, p. 80 commo ratio, p. 80 geometric series, p. 8 partial sum, p. 80 explicit rule, p. 87 recursive rule, p. 87 iteratio, p. 830 VOCABULARY EXERCISES. Copy ad complete: The values i the rage of a sequece are called the? of the sequece.. WRITING How ca you determie whether a sequece is arithmetic? 3. Copy ad complete: A()? rule gives as a fuctio of the term s positio umber i the sequece. 4. Copy ad complete: I a()? sequece, the ratio of ay term to the previous term is costat. REVIEW EXAMPLES AND EXERCISES Use the review examples ad exercises below to check your uderstadig of the cocepts you have leared i each lesso of Chapter.. Defie ad Use Sequeces ad Series pp E XAMPLE Fid the sum of the series a 4 3 a 4 0 a First term (i 4). Secod term Third term a Fourth term 4 The sum of the series is (i 4) EXERCISES EXAMPLES ad 6 o p. 797 for Exs. 8 Fid the sum of the series.. 6 ( 7) 6. 6 i (0 4i) 7. 7 i 8. k k 840 Chapter Sequeces ad Series

51 classzoe.com Chapter Review Practice. Aalyze Arithmetic Sequeces ad Series pp E XAMPLE Write a rule for the th term of the sequece 9, 3, 7,,,.... The sequece is arithmetic with first term a 9 ad commo differece d 4. So, a rule for the th term is: a ( )d Write geeral rule. 9 ( )(4) Substitute 9 for a ad 4 for d. 4 Simplify. EXERCISES EXAMPLES, 3, 4, ad o pp for Exs. 9 6 Write a rule for the th term of the arithmetic sequece. 9. 8,,,, 4, d 7, a 8 4. a 4 7, a 69 Fid the sum of the series.. (3 i) 3. 6 ( 3i) 4. (6i ). 30 (84 8i) 6. COMPUTER Joe buys a $600 computer o layaway by makig a $00 dow paymet ad the payig $ per moth. Write a rule for the total amout of moey paid o the computer after moths..3 Aalyze Geometric Sequeces ad Series pp E XAMPLE Fid the sum of the series 7 (3) i. The series is geometric with first term a ad commo ratio r 3. S 7 a r7 r Write rule for S Substitute for a ad 3 for r. 46 Simplify. EXERCISES EXAMPLES, 3, 4, ad o pp for Exs. 7 3 Write a rule for the th term of the geometric sequece. 7. 6, 64, 6, 4,, r, a a 44, a 3 6 Fid the sum of the series () i. 9 8() i. 3 i i Chapter Review 84

52 Fid.4 CHAPTER REVIEW Sums of Ifiite Geometric Series pp E XAMPLE Fid the sum of the series 4 i, if it exists. For this series, a ad r 4. Because r <, the sum of this series exists. a The sum is S r 4. EXERCISES EXAMPLES ad o pp. 8 8 for Exs. 4 3 Fid the sum of the ifiite geometric series, if it exists i i 6. Write the repeatig decimal as a fractio i lowest terms. 4(.3) i 7. 0.(0.) i Use Recursive Rules with Sequeces ad Fuctios pp E XAMPLE Write a recursive rule for the sequece 6, 0, 4, 8,,.... The sequece is arithmetic with first term a 6 ad commo differece d d Geeral recursive rule for 4 Substitute 4 for d. So, a recursive rule for the sequece is a 6, 4. EXERCISES EXAMPLES,, ad 3 o pp for Exs Write the first five terms of the sequece. 3. a 4, a 8, 34. a, p Write a recursive rule for the sequece. 3. 6, 8, 4, 6, 486, , 6, 9, 3, 8, , 3, 9,, 3, POPULATION A tow s populatio icreases at a rate of about % per year. I 000, the tow had a populatio of 6,000. Write a recursive rule for the tow s populatio P i year. Let represet Chapter Sequeces ad Series

53 Tell CHAPTER TEST whether the sequece is arithmetic, geometric, or either. Explai.., 9, 3, 7,.... 3, 6,, 4, , 0,,, , 7,, 9,... 8 Write the first six terms of the sequece a 4 8. a 6 Write the ext term of the sequece, ad the write a rule for the th term. 9.,, 7, 3, ,, 7, 37,.... 6, 7 0, 8, 9,.....6, 3., 4.8, 6.4,... 0 Fid the sum of the series i (4i 9) 6. 9 (i ) 7. 9() i i i i Write the repeatig decimal as a fractio i lowest terms Write a recursive rule for the sequece..,, 7, 43, , 0, 7, 4, , 4,,,... 8., 3, 9, 7,... Fid the first three iterates of the fuctio for the give iitial value. 9. f(x) 3x 7, x f(x) 8 x, x 0 3. f(x) x, x 0 3. QUILTS Use the patter of checkerboard quilts show.,, 3, 4, 8 a. What does represet for each quilt? What does represet? b. Make a table that shows ad for,, 3, 4,, 6, 7, ad 8. c. Use the rule 4 [ () ] to fid for,, 3, 4,, 6, 7, ad 8. Compare these values with the results i your table. What ca you coclude about the sequece defied by this rule? 33. AUDITIONS Several rouds of auditios are beig held to cast the three mai parts i a play. There are 307 actors at the first roud of auditios. I each successive roud of auditios, oe fourth of the actors from the previous roud remai. Fid a rule for the umber of actors i the th roud of auditios. For what values of does your rule make sese? Chapter Test 843

54 Stadardized TEST PREPARATION CONTEXT-BASED MULTIPLE CHOICE QUESTIONS Some of the iformatio you eed to solve a cotext-based multiple choice questio may appear i a table, a diagram, or a graph. P ROBLEM The frequecies (i hertz) of the otes o a piao form a geometric sequece. The frequecies of G (labeled 8 ) ad A (labeled 0 ) are show i the diagram. What is the approximate frequecy of E flat (labeled 4 )? A 47 Hz C 330 Hz B 3 Hz D 4 Hz 39 Hz 440 Hz Pla INTERPRET THE DIAGRAM The diagram gives you the frequecies of the 8th ad 0th otes. Use these frequecies to fid the frequecy of the 4th ote. STEP Write a system of equatios. Solutio Let be the frequecy (i hertz) of the th ote. Because the frequecies form a geometric sequece, a rule for has the form a r. From the diagram, a 8 39 ad a Use these values to write a system of equatios. a 8 a r 8 39 a r 7 Equatio STEP Solve the system of equatios to fid the values of r ad a. a 0 a r a r 9 Equatio a 39 r 7 Solve Equatio for a r p r 9 7 Substitute 39 r for a 7 i Equatio r Simplify.. ø r Divide each side by ø r Take positive square root of each side. Fid a by substitutig the value of r ito revised Equatio. STEP 3 Write a rule for the th term ad fid a 4. a 39 r 39 7 (.06) ø 6 7 A rule for the sequece is a r 6(.06). So, a 4 6(.06) 3 ø 3. c The correct aswer is B. A B C D 844 Chapter Sequeces ad Series

55 P ROBLEM The first 4 terms of a ifiite arithmetic sequece are show i the graph. Which rule describes the th term i the sequece? A B C D (4, 3) (3, ) (, ) (, 3) Pla INTERPRET THE GRAPH I order to fid a rule for the sequece, you must first use the graph to write the terms of the sequece. STEP Write the terms of the sequece. STEP Fid the first term ad the commo differece. STEP 3 Write a rule for the th term. Solutio The poits show i the graph are: (, 3), (, ), (3, ), (4, 3) Therefore, the sequece is 3,,, 3,.... The first term a of the sequece is 3. Because each term after the first is more tha the previous term, the commo differece d is. a ( )d Write geeral rule for a arithmetic sequece. 3 ( ) Substitute 3 for a ad for d. 3 Distributive property Simplify. c The correct aswer is A. A B C D PRACTICE I Exercises ad, use the graph i Problem.. What is the value of a? A 3 B C 30 D 6. Which statemet is true about the sequece that is graphed? A The sum of the first 4 terms is 40. B The value of a 0 is 40. C A recursive rule for the sequece is a,. D The ratio of ay term to the previous term is costat. Stadardized Test Preparatio 84

56 Stadardized TEST PRACTICE MULTIPLE CHOICE. The diagram shows the bouce heights of a basketball ad a baseball dropped from a height of 0 feet. O each bouce, the basketball bouces to 36% of its previous height, ad the baseball bouces to 30% of its previous height. About how much greater is the total distace traveled by the basketball tha the total distace traveled by the baseball? 0 ft 0 ft I Exercises 4 ad, use the iformatio below. Cheryl is researchig her lieage for a history project. So far, she has created a family tree for three geeratios, as show below. Cheryl is oly icludig relatives from whom she is directly desceded. Sibligs are ot icluded. Materal gradmother Mother Materal gradfather Pateral gradmother Father Pateral gradfather Cheryl 3.6 ft 3.6 ft Basketball A.34 feet C.6 feet.3 ft.3 ft 3 ft 3 ft Baseball B.00 feet D.63 feet 0.9 ft 0.9 ft. The table shows the domai ad rage of a sequece. Which recursive rule describes the sequece? Domai 3 4 Rage Assume that Cheryl is i geeratio, her parets are i geeratio, ad so o. Let be the umber of relatives i geeratio. What is a rule for? A B C D. Cheryl creates a family tree with 8 geeratios of her family. How may people are i her family tree? A 8 B 64 C 8 D I Exercises 6 ad 7, use the diagram of a stack of blocks. A a 0, 0 B a 0, 0 C a 0, 0. D a 0, 3. What type of sequece is graphed at the right? 6. Which rule describes the umber of blocks i the th layer, where represets the top layer? A B ( ) A Arithmetic C D B Geometric with 0 < r < C Geometric with r > D Neither arithmetic or geometric 3 7. Which sum gives the umber of blocks show? A C 0 i 0 i (i ) B i(i ) D 4 4 (i ) i(i )