Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4


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1 3330_0903qxd /5/05 :3 AM Page 663 Sectio Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial sums of geometric sequeces Fid the sum of a ifiite geometric series Use geometric sequeces to model ad solve reallife problems Why you should lear it Geometric sequeces ca be used to model ad solve reallife problems For istace, i Exercise 99 o page 670, you will use a geometric sequece to model the populatio of Chia Geometric Sequeces I Sectio 9, you leared that a sequece whose cosecutive terms have a commo differece is a arithmetic sequece I this sectio, you will study aother importat type of sequece called a geometric sequece Cosecutive terms of a geometric sequece have a commo ratio Defiitio of Geometric Sequece A sequece is geometric if the ratios of cosecutive terms are the same So, the sequece a, a, a3, a4,, a is geometric if there is a umber r such that a r, a a3 r, a a4 r, a3 r 0 ad so the umber r is the commo ratio of the sequece Example Examples of Geometric Sequeces a The sequece whose th term is is geometric For this sequece, the commo ratio of cosecutive terms is, 4, 8, 6,,, 4 Bob Krist/Corbis Begi with b The sequece whose th term is 4 3 is geometric For this sequece, the commo ratio of cosecutive terms is 3, 36, 08, 34,, 4 3, Begi with 36 3 c The sequece whose th term is 3 is geometric For this sequece, the commo ratio of cosecutive terms is 3,,,,,, Begi with Now try Exercise The sequece, 4, 9, 6,, whose th term is, is ot geometric The ratio of the secod term to the first term is a 4 4 a a 9 but the ratio of the third term to the secod term is 3 a 4
2 3330_0903qxd /5/05 :3 AM Page Chapter 9 Sequeces, Series, ad Probability I Example, otice that each of the geometric sequeces has a th term that is of the form ar, where the commo ratio of the sequece is r A geometric sequece may be thought of as a expoetial fuctio whose domai is the set of atural umbers The th Term of a Geometric Sequece The th term of a geometric sequece has the form a a r where r is the commo ratio of cosecutive terms of the sequece So, every geometric sequece ca be writte i the followig form a, a, a 3, a 4, a 5,, a, a a a a r 4 r 3 r, a r,,,,, a r, Additioal Example Write the first five terms of the geometric sequece whose first term is a 9 ad whose commo ratio is r 3 9, 3,, a 3, 9 FIGURE If you kow the th term of a geometric sequece, you ca fid the th term by multiplyig by r That is, a ra Example Fidig the Terms of a Geometric Sequece Write the first five terms of the geometric sequece whose first term is a 3 ad whose commo ratio is r The graph the terms o a set of coordiate axes Startig with 3, repeatedly multiply by to obtai the followig a 3 st term a 3 6 d term a 3 3 3rd term a th term a th term Figure 95 shows the first five terms of this geometric sequece Now try Exercise Example 3 Fidig a Term of a Geometric Sequece Fid the 5th term of the geometric sequece whose first term is 0 ad whose commo ratio is 05 a 5 a r Formula for geometric sequece Substitute 0 for a, 05 for r, ad 5 for Use a calculator Now try Exercise 7
3 3330_0903qxd /5/05 :3 AM Page 665 Sectio 93 Geometric Sequeces ad Series 665 Example 4 Fidig a Term of a Geometric Sequece Fid the th term of the geometric sequece 5, 5, 45, The commo ratio of this sequece is r Because the first term is a 5, you ca determie the th term to be a a r a 53 Formula for geometric sequece Substitute 5 for a, 3 for r, ad for 577,47 Use a calculator 885,735 Simplify Now try Exercise 35 If you kow ay two terms of a geometric sequece, you ca use that iformatio to fid a formula for the th term of the sequece Example 5 Fidig a Term of a Geometric Sequece Remember that r is the commo ratio of cosecutive terms of a geometric sequece So, i Example 5, a 0 a r 9 a r r r r 6 a a a a 3 a a 4 a 3 r 6 a 4 r 6 The fourth term of a geometric sequece is 5, ad the 0th term is 564 Fid the 4th term (Assume that the terms of the sequece are positive) The 0th term is related to the fourth term by the equatio a 0 a 4 r 6 Multiply 4th term by r 04 Because a ad a 4 5, you ca solve for r as follows r6 64 r 6 r Substitute for ad 5 for a 4 Divide each side by 5 Take the sixth root of each side You ca obtai the 4th term by multiplyig the 0th term by r 4 a Multiply the 0th term by r 40 4 a 0 r Substitute for ad for r Simplify Now try Exercise a 0 a 0
4 3330_0903qxd /5/05 :3 AM Page Chapter 9 Sequeces, Series, ad Probability The Sum of a Fiite Geometric Sequece The formula for the sum of a fiite geometric sequece is as follows The Sum of a Fiite Geometric Sequece The sum of the fiite geometric sequece a, a r, a r, a r 3, a r 4,, a r with commo ratio is give by S r a r a r r For a proof of the sum of a fiite geometric sequece, see Proofs i Mathematics o page 73 Example 6 Fidig the Sum of a Fiite Geometric Sequece Fid the sum 403 By writig out a few terms, you have Now, because a 4, r 03, ad, you ca apply the formula for the sum of a fiite geometric sequece to obtai Now try Exercise 57 Formula for the sum of a sequece Substitute 4 for a, 03 for r, ad for Use a calculator Whe usig the formula for the sum of a fiite geometric sequece, be careful to check that the sum is of the form a r S a r r Expoet for r is i If the sum is ot of this form, you must adjust the formula For istace, if the sum i Example 6 were 403 i, the you would evaluate the sum as follows 403 i a 403, r 03,
5 3330_0903qxd /5/05 :3 AM Page 667 Sectio 93 Geometric Sequeces ad Series 667 Exploratio Use a graphig utility to graph y r x r for r, 3, ad 5 What happes as x? Use a graphig utility to graph for r 5,, ad 3 What happes as x? 4 y r x r Geometric Series The summatio of the terms of a ifiite geometric sequece is called a ifiite geometric series or simply a geometric series The formula for the sum of a fiite geometric sequece ca, depedig o the value of r, be exteded to produce a formula for the sum of a ifiite geometric series Specifically, if the commo ratio r has the property that r <, it ca be show that r becomes arbitrarily close to zero as icreases without boud Cosequetly, a r r a 0 r This result is summarized as follows The Sum of a Ifiite Geometric Series If r <, has the sum S i0 the ifiite geometric series a a r a r a r 3 a r a r i a r as Note that if r, the series does ot have a sum Example 7 Fidig the Sum of a Ifiite Geometric Series Ifiite geometric series ad their sums have importat uses i calculus Refer to Exercise 0 i this sectio Activities Determie which of the followig are geometric sequeces (a) 3, 6, 9,, 5, (b), 4, 8, 6, 3, (c),,,,, (d) 4,,,, 4, (e), 4, 6, 64, 56, Aswer: (b), (c), (d) Fid the sum 0 6 Aswer: Fid the sum 6 Aswer: 3 Fid each sum a 406 b a b Now try Exercise a r a r
6 3330_0903qxd /5/05 :3 AM Page Chapter 9 Sequeces, Series, ad Probability Applicatio Example 8 Icreasig Auity Recall from Sectio 3 that the formula for compoud iterest is A P r t So, i Example 8, $50 is the pricipal P, 006 is the iterest rate r, is the umber of compoudigs per year, ad is the time t i years If you substitute these values ito the formula, you obtai A A deposit of $50 is made o the first day of each moth i a savigs accout that pays 6% compouded mothly What is the balace at the ed of years? (This type of savigs pla is called a icreasig auity) The first deposit will gai iterest for 4 moths, ad its balace will be A The secod deposit will gai iterest for 3 moths, ad its balace will be A The last deposit will gai iterest for oly moth, ad its balace will be A The total balace i the auity will be the sum of the balaces of the 4 deposits Usig the formula for the sum of a fiite geometric sequece, with A ad r 005, you have S $7796 Now try Exercise 07 Substitute for A, 005 for r, ad 4 for Simplify Writig About Mathematics Suggestio: For the sake of simplicity, you may wat to cosider supplyig each group with the tools they eed for this activity rather tha havig them supply the materials themselves W RITING ABOUT MATHEMATICS A Experimet You will eed a piece of strig or yar, a pair of scissors, ad a tape measure Measure out ay legth of strig at least 5 feet log Double over the strig ad cut it i half Take oe of the resultig halves, double it over, ad cut it i half Cotiue this process util you are o loger able to cut a legth of strig i half How may cuts were you able to make? Costruct a sequece of the resultig strig legths after each cut, startig with the origial legth of the strig Fid a formula for the th term of this sequece How may cuts could you theoretically make? Discuss why you were ot able to make that may cuts
7 3330_0903qxd /5/05 :3 AM Page 669 Sectio 93 Geometric Sequeces ad Series Exercises VOCABULARY CHECK: Fill i the blaks A sequece is called a sequece if the ratios betwee cosecutive terms are the same This ratio is called the ratio The th term of a geometric sequece has the form 3 The formula for the sum of a fiite geometric sequece is give by 4 The sum of the terms of a ifiite geometric sequece is called a 5 The formula for the sum of a ifiite geometric series is give by PREREQUISITE SKILLS REVIEW: Practice ad review algebra skills eeded for this sectio at wwweduspacecom I Exercises 0, determie whether the sequece is geometric If so, fid the commo ratio 5, 5, 45, 35, 3,, 48, 9, 3 3,,, 30, 4 36, 7, 8, 9, 5,, 4, 8, 6 5,, 0, 004, 7 8 8, 4,,, 8 9, 6, 4, 3 9, 0 5, 7, 3 9, 4, 3, 4,, I Exercises 0, write the first five terms of the geometric sequece a, r 3 a 6, r 3 a, r 4 a, r 3 5 a 5, r 0 6 a 6, r 4 7 a, r e 8 a 3, r 5 I Exercises 35 4, fid the idicated th term of the geometric sequece 35 9th term: 7,, 63, 36 7th term: 3, 36, 43, 37 0th term: 5, 30, 80, 38 d term: 4, 8, 6, 39 3rd term: a a 4 7 6, 4 40 st term: a a 5 3 3, th term: a 4 8, a th term: a a , 7 I Exercises 43 46, match the geometric sequece with its graph [The graphs are labeled (a), (b), (c), ad (d)] (a) a (b) a 9 a, r x 0 a 5, r x 4 I Exercises 6, write the first five terms of the geometric sequece Determie the commo ratio ad write the th term of the sequece as a fuctio of a 64, a k a k a 8, a k 3a k 3 a 7, a k a k 4 a 5, a k a k 5 a 6, a k 3 a k 6 a 48, a k a k I Exercises 7 34, write a expressio for the th term of the geometric sequece The fid the idicated term 7 a 3 4, r, 0 8 a 5, r, 8 9 6, r 3, 30 a 64, r 4, a 00, r e x, 9 a, r 3, 8 33 a 500, r 0, a 000, r 005, 60 (c) a (d) 43 a a a 46 a a
8 3330_0903qxd /5/05 :3 AM Page Chapter 9 Sequeces, Series, ad Probability I Exercises 47 5, use a graphig utility to graph the first 0 terms of the sequece 47 a a a a 05 5 a 3 5 a 0 I Exercises 53 7, fid the sum of the fiite geometric sequece I Exercises 73 78, use summatio otatio to write the sum I Exercises 79 9, fid the sum of the ifiite geometric series i i I Exercises 93 96, fid the ratioal umber represetatio of the repeatig decimal Graphical Reasoig I Exercises 97 ad 98, use a graphig utility to graph the fuctio Idetify the horizotal asymptote of the graph ad determie its relatioship to the sum f x 6 x 05, f x x 08, 0 5 Model It 99 Data Aalysis: Populatio The table shows the populatio a of Chia (i millios) from 998 through 004 (Source: US Cesus Bureau) Year Populatio, (a) Use the expoetial regressio feature of a graphig utility to fid a geometric sequece that models the data Let represet the year, with 8 correspodig to 998 (b) Use the sequece from part (a) to describe the rate at which the populatio of Chia is growig a
9 3330_0903qxd /5/05 :3 AM Page 67 Sectio 93 Geometric Sequeces ad Series 67 Model It (cotiued) (c) Use the sequece from part (a) to predict the populatio of Chia i 00 The US Cesus Bureau predicts the populatio of Chia will be 3746 millio i 00 How does this value compare with your predictio? (d) Use the sequece from part (a) to determie whe the populatio of Chia will reach 3 billio 00 Compoud Iterest A pricipal of $000 is ivested at 6% iterest Fid the amout after 0 years if the iterest is compouded (a) aually, (b) semiaually, (c) quarterly, (d) mothly, ad (e) daily 0 Compoud Iterest A pricipal of $500 is ivested at % iterest Fid the amout after 0 years if the iterest is compouded (a) aually, (b) semiaually, (c) quarterly, (d) mothly, ad (e) daily 0 Depreciatio A tool ad die compay buys a machie for $35,000 ad it depreciates at a rate of 30% per year (I other words, at the ed of each year the depreciated value is 70% of what it was at the begiig of the year) Fid the depreciated value of the machie after 5 full years 03 Auities A deposit of $00 is made at the begiig of each moth i a accout that pays 6%, compouded mothly The balace A i the accout at the ed of 5 years is A Fid A 04 Auities A deposit of $50 is made at the begiig of each moth i a accout that pays 8%, compouded mothly The balace A i the accout at the ed of 5 years is A Fid A 05 Auities A deposit of P dollars is made at the begiig of each moth i a accout earig a aual iterest rate r, compouded mothly The balace A after t years is A P r P r Show that the balace is A P r t r P r t 06 Auities A deposit of P dollars is made at the begiig of each moth i a accout earig a aual iterest rate r, compouded cotiuously The balace after years is A Pe r Pe r A t Pe tr Show that the balace is Auities I Exercises 07 0, cosider makig mothly deposits of P dollars i a savigs accout earig a aual iterest rate r Use the results of Exercises 05 ad 06 to fid the balace A after t years if the iterest is compouded (a) mothly ad (b) cotiuously A Per e rt e r P $50, r 7%, t 0 years P $75, r 3%, t 5 years P $00, r 0%, t 40 years P $0, r 6%, t 50 years Auities Cosider a iitial deposit of P dollars i a accout earig a aual iterest rate r, compouded mothly At the ed of each moth, a withdrawal of W dollars will occur ad the accout will be depleted i t years The amout of the iitial deposit required is P W r W r Show that the iitial deposit is P W r r t W r t Auities Determie the amout required i a retiremet accout for a idividual who retires at age 65 ad wats a icome of $000 from the accout each moth for 0 years Use the result of Exercise ad assume that the accout ears 9% compouded mothly Multiplier Effect I Exercises 3 6, use the followig iformatio A tax rebate has bee give to property owers by the state govermet with the aticipatio that each property ower speds approximately p% of the rebate, ad i tur each recipiet of this amout speds p% of what they receive, ad so o Ecoomists refer to this exchage of moey ad its circulatio withi the ecoomy as the multiplier effect The multiplier effect operates o the idea that the expeditures of oe idividual become the icome of aother idividual For the give tax rebate, fid the total amout put back ito the state s ecoomy, if this effect cotiues without ed Tax rebate p% 3 $400 75% 4 $50 80% 5 $600 75% 6 $ %
10 3330_0903qxd /5/05 :3 AM Page Chapter 9 Sequeces, Series, ad Probability 7 Geometry The sides of a square are 6 iches i legth A ew square is formed by coectig the midpoits of the sides of the origial square, ad two of the resultig triagles are shaded (see figure) If this process is repeated five more times, determie the total area of the shaded regio You ca fid the th term of a geometric sequece by multiplyig its commo ratio by the first term of the sequece raised to the th power 3 Writig Write a brief paragraph explaiig why the terms of a geometric sequece decrease i magitude whe < r < 4 Fid two differet geometric series with sums of 4 Skills Review 8 Sales The aual sales a (i millios of dollars) for Urba Outfitters for 994 through 003 ca be approximated by the model a 546e 07, where represets the year, with 4 correspodig to 994 Use this model ad the formula for the sum of a fiite geometric sequece to approximate the total sales eared durig this 0year period (Source: Urba Outfitters Ic) 9 Salary A ivestmet firm has a job opeig with a salary of $30,000 for the first year Suppose that durig the ext 39 years, there is a 5% raise each year Fid the total compesatio over the 40year period 0 Distace A ball is dropped from a height of 6 feet Each time it drops h feet, it rebouds 08h feet (a) Fid the total vertical distace traveled by the ball (b) The ball takes the followig times (i secods) for each fall Sythesis s 6t 6, s 6t 608, s 3 6t 608, s 4 6t 608 3, s 6t 608, Begiig with s, the ball takes the same amout of time to bouce up as it does to fall, ad so the total time elapsed before it comes to rest is t Fid this total time 4, 5,, 3 09 s 0 if t s 0 if t 09 s 3 0 if t 09 s 4 0 if t 09 3 s 0 if t 09 True or False? I Exercises ad, determie whether the statemet is true or false Justify your aswer A sequece is geometric if the ratios of cosecutive differeces of cosecutive terms are the same I Exercises 5 8, evaluate the fuctio for fx 3x ad gx x gx f x f gx g f x I Exercises 9 3, completely factor the expressio 9 9x 3 64x 30 x 4x x 3x 5 3 6x 4x 4 I Exercises 33 38, perform the idicated operatio(s) ad simplify xx 3 x 3 x 3 x xx 7 x 7 6xx x 3x 3 6x 3 x 5 0 x x 3 3 x 5 7 x x 8 x x 4 4 x x 4 x x 4 39 Make a Decisio To work a exteded applicatio aalyzig the amouts spet o research ad developmet i the Uited States from 980 to 003, visit this text s website at collegehmcocom (Data Source: US Cesus Bureau)
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