44 Chapter 3. Find the 13 term and the sum of the first 9 terms of the geometric sequence 48, 24, 12, 6, 3, 3 2 Solution 2

Transcription

1 44 Chapter 3 Fd e 3 ter ad e su of e frst 9 ters of e geoetrc sequece 48, 24, 2, 6, 3, 3 2, á. We have a 48 ad r 2. Usg part (a) of Theore 3.2, we fd at e 3 ter s 48( 2 ) Usg (3.4d), e su of e frst 9 ( ) 9 ters s ( ) BASIC RESULTS Joh borrows 500 fro a face copay ad wshes to pay t back w equal aual payets at e ed of each of e ext te years. If.7, what should hs aual payet be? Jacta buys a house ad takes out a 50,000 ortgage. If e ortgage rate s 3% covertble seaually, what should her oly payet be to pay off e ortgage 20 years? Elee deposts 2000 a bak accout every year for years. If. 06, how uch has she accuulated at e te of e last depost? All of ese questos have oe g coo: ey volve a seres of payets ade at regular tervals. Such a seres of payets s called a auty. I e ree cases above, e payets are of equal aout, ad at wll be e case w all autes studed s secto. Later, however, we wll study ore geeral autes. Autes tur up ay dfferet types of facal trasactos. Fro e pot of vew of practcal applcatos, a coplete uderstadg of autes s a absolute ust! We shall start by cosderg a auty uder whch payets of are ade at e ed of each perod for perods. Soetes a perod wll be oe year, as w Joh s loa above, but oer perods are certaly possble. It wll be assued roughout at, as w Joh s loa, e terest perod ad e payet perod are equal. Whe s s ot e case, as w Jacta s ortgage, for exaple, we wll frst fd e equvalet rate of terest per payet perod ad e proceed w our soluto. Level payets of a aout oer a ca be hadled by ultplyg by e aout of e payet, as we shall see e exaples.

2 Autes 45 a s Æ Æ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ââ FIGURE 3. A te dagra showg payets of s gve Fgure 3.. The preset value of s auty at te 0 s deoted by a. The accuulated value of s auty at te s deoted by s. We shall ow derve a forula for a. Takg e value at te 0 of each of e payets tur, we obta a vv v âv. (3. 5) 2 3 Ths s e su of ters of a geoetrc sequece w a v ad r v. Usg Forula (3.4d) developed Secto 3., we obta a v( v ) v v v v v. (3. 6) Forula (3.6) s crucal, ad wll be used frequetly roughout e rest of e text. It s easy ow to get a forula for s. Sce s s e value of e sae auty years after a has bee calculated, t follows at s a ( ) v ( ) ( ) v ( ) ( ). (3.7)

3 46 Chapter 3 Let us edately proceed to soe practcal exaples. Exaple 3.3 Fd Joh s payet e proble stated e frst paragraph of s secto. X X X qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ 0 FIGURE 3.2 Let e payet be X. Sce e preset value of 0 payets of s a, e preset value of 0 payets of X wll be X a. Thus we have The (. 7) X a0 X a v 0 0 ( 7). Exaple 3.4 Fd e accuulated value Elee s bak accout e proble stated e rd paragraph of s secto qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ X FIGURE 3.3 Sce each depost s 2000, e accuulated value wll be gve drectly (.06) by X 2000s , Exaple 3.5 Fd Jacta s ortgage payet e proble stated e secod paragraph of s secto. X X X qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ ,000 FIGURE 3.4

4 Autes 47 As etoed earler, we frst have to fd e effectve oly rate of terest equvalet to 3% covertble seaually. Ths s because our forulae for a ad s are based o e assupto at e terest perod ad payet perod are e sae. Lettg s oly rate be j, we have j.3 2 /6. Now we let e ortgage payet be X. Note at ere are 240 oly payets e 20-year ter of e ortgage, so we have X 50,000 a 50,000 ad X j v.. Exaple 3.6 Elroy takes out a loa of $5000 to buy a car. No payets are due for e frst 8 os, but begg w e ed of e 9 o, he ust ake 60 equal oly payets. If.8, fd (a) e aout of each payet; (b) e aout of each payet f ere s o payet-free perod, (.e., f e frst payet s due oe o ad e reag 59 are ade o a oly bass ereafter). (a) We frst ote at a oly rate of terest j s requred. Sce ( j) 2.8, we obta j (.8) /2. Let e aout of each payet be X Æ \ \ \ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 ââ ââ 68 FIGURE 3.5 We ow observe at s does ot ft to e stadard auty patter, sce X a60 wll gve us e value of e payets at o 8, oe o before e frst payet. The value of e loa at te 8 s 5,000( j) 8, sce t wll accrue terest for eght os, eve ough o payets are requred. Thus we have e ( j) equato of value X a ( j), so at X a ( j) j Evaluatg a60, X v

5 48 Chapter 3 (b) I s case, we just have X a , whch we solve for 5000j X Ths shows at, over e ext 5 ( j) years, e total aout of extra oey pad for postpog e frst payet for 8 os wll be ! The prevous exaple beautfully deostrates e power of our calculators. I pre-calculator days e evaluato of e ter 60 j a60 ( j ) would have caused serous dffcultes. Eve terest tables would ot have helped, because e terest rate j s ot oe for whch tables were costructed. All we do ow, however, s press a few buttos ( e rght order) ad e aswer appears! Exaple 3.7 (a) Prove e detty a v. (b) Gve a verbal terpretato of s detty. (a) a v, so a v, ad a v, as requred. (b) The ter a ca be ought of as e preset value of a auty w level payet at e ed of each year for years (see Fgure 3.6). The ter v s e preset value of at year. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ Å FIGURE 3.6 Iage vestg at te 0. At e ed of e frst year, e terest s separated off fro e orgal vestet, ad e aout of e vestet s back to. Ths procedure cotues for years, leavg at e ed of years ad e auty of whch was reoved each year. The preset value of ese ters s v ad a, respectvely. There are two oer sybols coo usage w autes, aely a ad s.

6 Autes 49 ä s e preset value of e auty descrbed earler at e te of e frst payet, ad s s e accuulated value oe year after e last payet has bee ade. Our four fuctos, a, a, s ad s are llustrated Fgure 3.7. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ a ä s s FIGURE 3.7 Maeatcally, ere s og very exctg gog o here. We ca see edately fro Fgure 3.7 at a a ( ), ad at s s ( ). These relatoshps lead to forulae for a ad s at are aalogous to Forulae (3.6) ad (3.7) for a ad s, respectvely. We have a a ( ) v v ( ) v d, (3. 8) where d s e effectve rate of dscout defed Chapter. Slarly, e reader should show at s ( ) d. (3. 9) Observe at ä ca also be descrbed as e preset value of payets of ade at e begg of each perod for perods, ad s ca be descrbed as e accuulated value of e sae payets at e ed of e last perod. Sce e payets are at e beggs of e perods, t follows at s s er accuulated value a full perod after e last payet.

7 50 Chapter 3 There are ay dettes relatg e four quattes we have troduced. I addto to e oes etoed earler, we ote at s a ( ) (3.0) ad da v, (3. ) bo of whch have ce verbal terpretatos. Oer relatoshps wll be preseted as exercses. Let us do a exaple. Iage at Hery takes out a loa of 000 ad repays t w 0 equal yearly payets, e frst oe due at e te of e loa. I s case, f X s e aout of each payet, a approprate equato of value would be X ä (3.2a) X X X X qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ FIGURE 3.8 We use a 0 here because e auty sybol a 0 assues we are takg e preset value oe year before e frst payet, ad at s ot e case w Hery s loa. A equally good equato of value would be X a 000 v, (3. 2b) 0 because 000 v would be e value of e loa oe year earler, whch allows us to use our frst auty sybol. Oce e rate of terest s kow, we ca fd e aout of Hery s loa payet. Let us observe as well at a rd acceptable equato of value for Hery s proble would be Exaple 3.8 X( a ) 000. (3.2c) 9 Usg all ree equatos of value, fd Hery s loa payet f.6.

8 Autes 5 (a) Frst we cosder Equato (3.2a), whch s X ä The X 000 d ä To use s approach we eed to fd 0 v d Ths leads to e aswer X (b) Equato (3.2b) states at X a 000 v. The we have 000 Š 6. (. 6) X Š 6. (c) Fally, cosder Equato (3.2c), whch gves us e equato X a v The oral of Exaple 3.8 s at ere s ore a oe way to work out s kd of proble. However, we reterate e portace of keepg as ay decal places as possble durg your calculatos. I ay textbooks, e ter auty-edate s used for e case where payets are ade at e ed of e perod, ad auty-due s used whe payets are ade at e begg of e perod. As we have just llustrated, however, e sae techques ca be used bo cases. The followg exaple llustrates at ere are ay possble ways of aalyzg autes. Exaple 3.9 Cosder a auty whch pays at e begg of each year for years. Expla verbally why each of e followg expressos gves e curret value of s auty at e ed of year. (See Fgure 3.9 o e followg page.) (a) a ( ) (b) a ( ) (c) s v (d) s v (e) s a (f) s a (g) s a 0

9 52 Chapter 3 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 2 ââ ââ Å FIGURE 3.9 I e te dagra for s auty Fgure 3.9, we have deoted by Å e pot at whch we wat e value of e auty. (a) a s e value at year. To get to Å, we ust ove years to e future, so we have a ( ). (b) a s e value at year 0. So a ( ) oves us years to e future. (c) s s e value at year. Hece we ove back years. (d) s s e value at year, so we ove back years. (e) s s e value of e frst payets at te, ad a s e value of e last payets at te. (f) s s e value of e frst payets at te, ad a s e value of e last payets at e sae te. (g) Here e sgle payet of at te s separated off fro e ä part (f), leavg. a The above exaple should llustrate how careful we ust be whe workg w ese fuctos, but also at we have cosderable flexblty usg e to express a auty value at soe pot of te. 3.3 PERPETUITIES A perpetuty s a auty whose payets cotue forever. The te dagra s show Fgure 3.0 below. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ââ Å a _ FIGURE 3.0

10 Autes 53 The value of s auty oe year before e frst payet s a _. We have a l a _ Ä_ l v Ä_, (3.3) sce l v 0, as log as 0. Ä_ We ca see verbally why Forula (3.3) should be true: f a prcpal of s vested at rate, e e terest Š ca be reoved at e ed of each year, leavg e orgal prcpal tact forever. As Secto 3.2, e sybol ä _ represets e value of a perpetuty at e te of e frst payet. The followg dettes are left as exercses for e reader: a a ( ), (3.4) _ _ ad a a, (3.5) _ _ a _ d. (3. 6) 3.4 UNKNOWN TIME AND UNKNOWN RATE OF INTEREST We wll cosder here several exaples volvg autes where e leg of te or e rate of terest volved s e ukow. Exaple 3.0 A fud of 5000 s used to award scholarshps of aout 500, oe per year, at e ed of each year for as log as possble. If.09, fd e uber of scholarshps whch ca be awarded, ad e aout left e fud oe year after e last scholarshp has bee awarded.

11 66 Chapter 3 (a) The aretc sequece 2, 7, 2, 7, á. (b) The aretc sequece w a 7 ad d 3. (c) The aretc sequece whose 5 ter s 9 ad whose 9 ter s 47. (d) The geoetrc sequece 5, 5, 45, á. (e) The geoetrc sequece 3, 3, 3, , á. (f) The geoetrc sequece whose 5 ter s 2 9 ad whose 8 ter s Prove Theores 3.(a), 3.(b), 3.2(a) ad 3.2(b) usg aeatcal ducto. 3.2 Basc Results 3-3. Heretta borrows 6500 order to buy furture. She wshes to pay e loa back by eas of 2 aual payets, e frst to be ade oe year after e loa s take out. If.3, fd e aout of each payet Aswer Questo 3 f e loa s to be pad back w 44 oly payets, e frst oe due oe o after e loa s take out Alphose deposts 450 a bak accout at e begg of each year, startg 977 ad cotug for 20 years. If.08, fd e aout hs accout at e ed of A auty pays 000 a year for 8 years. If.08, fd each of e followg: (a) The value of e auty oe year before e frst payet. (b) The value of e auty oe year after e last payet. (c) The value of e auty at e te of e ff payet. (d) The uber of years e auty would have to ru order at ts curret preset value be doubled. (e) The uber of years e auty would have to ru order at ts curret preset value be trpled Prove each of e followg dettes: (a) a a v a (b) a a v s (c) s s ( ) s (d) s s ( ) a

12 Autes Gve verbal terpretatos for each of e dettes Questo Prove at a s Prove each of e followg dettes: (a) a a (b) s s 3-. Gve verbal terpretatos for e dettes Questo Rak, a ad s creasg order of agtude. Uder what codtos wll equalty hold for all? 3-3. Harret wshes to accuulate 85,000 a fud at e ed of 25 years. If she deposts 000 e fud at e ed of each of e frst 0 years, ad 000 x at e ed of each of e last 5 years, fd x f e fud ears 7% effectve. s2 s s Show at s s s Prove each of e followg dettes: (a) a a v (b) s s ( ) 3-6. Gve verbal terpretatos for e dettes Questo 5. b 3-7. Show at "( s s ) s s ( b a ). ta t t, A auty rus for 25 years as follows: at e ed of each of e frst te years 500 s pad, ad e at e ed of each of e last 5 years 300 s pad. If.08, fd e value of s auty ree years before e frst payet Edward buys a ew house ad takes out a ortgage of 60,000. To pay off e ortgage, he wll ake oly payets w e frst payet due oe o. Gve (2).2, fd e aout

13 68 Chapter 3 of hs payet f (a) e payets wll cotue for e ext 25 years; (b) e payets wll cotue for e ext 20 years; (c) e payets wll cotue for e ext 0 years Rework Questo 9 f e oal rate of terest covertble seaually s 6% stead of 2% A a wshes to accuulate a sall peso by depostg 2500 at e begg of each year for 25 years. Startg at e ed of e year whch e fal depost s ade, he wll ake 20 aual wdrawals. Fd e aout of each wdrawal, f.07 durg e frst 25 years ad. ereafter A seres of payets are ade as follows: at e ed of e frst year, 2 at e ed of each of e ext years, ad at e ed of year. Show at e value of ese payets at t 0s a ä Gve a verbal explaato of why e forula Questo 22 s correct A auty cossts of payets of, e frst to be ade at e ed of 7 years ad e oer payets to be ade at ree year tervals ereafter. Show at e preset value of e auty s a3 7 a7 a Albert Glover, star rd basea w e Blue Jays, s gve a choce of cotracts: (a) 3,200,000 per year for e ext fve years, payable at e ed of each year. (b) 3,000,000 per year for e ext fve years, payable at e begg of each year. (c),800,000 per year for e ext te years, payable at e ed of each year. If.04, fd e value of each of ese cotracts at e begg of e frst year. Repeat for.06.

14 Autes Fd e rage of terest rates for whch each of e cotracts Questo 25 has a hgher preset value e e oer two Cosder a auty where k payets of are ade, e frst occurrg k years fro ow w e payets cotug at k-year tervals ereafter, utl a perod of years has passed. Prove at a e preset value of ese payets s equal to Show at e accuulated value of e auty Questo 27 s edately after e last payet s Gve verbal terpretatos for e forulae Questo 27 ad Questo Prove at e preset value of a auty whch pays at e ed of each of a year for e ext years s equal to v. Ths ( ) ( ) preset value s deoted by a Prove at e accuulated value of e auty Questo 30 at ( ) e te of e last payet s ( ). Ths accuulated ( ) value s deoted by s Derve a expresso for e preset value of a auty uder whch payets are 2,, 2,, á at e ed of every year for e ext 25 years If a xad a y, express d as a fucto of x ad y A loa of 25,000 s to be repad by aual payets at e ed of each year for e ext 20 years. Durg e frst 5 years e payets are k per year; durg e secod 5 years e payets are 2k per year; durg e rd 5 years, 3 k per year; ad durg e four 5 years, 4 k per year. If.2, fd k. s k Gve a 2 ad a 2, fd a. 2 4 s k

15 70 Chapter Gve ät ad ät 9.499, fd e effectve rate of terest A jured worker subts a Workers Copesato cla. It s decded at she s ettled to aual edcal payets of 20,000 for e ext 0 years ad equal aual dety payets for e ext 20 years. The edcal payets wll beg edately, ad e dety payets wll beg oe year s te. The surace copay has establshed a fud of 680,000 to support ese payets. Fd e aout of each aual dety payet assug Perpetutes Prove dettes (3.4), (3.5) ad (3.6) Gve.5, fd e preset value of a auty of 00 per year cotug forever f (a) e frst payet s due oe year; (b) e frst payet s due edately; (c) e frst payet s due 5 years A perpetuty of 500 per year, w e frst payet due oe year hece, s wor Fd Deposts of 000 are placed to a fud at e ed of each year for e ext 25 years. Fve years after e last depost, aual payets coece ad cotue forever. If.09, fd e aout of each payet A loa of 5000 s repad by aual payets cotug forever, e frst oe due oe year after e loa s take out. If e payets are X, 2 X, X, 2 X, á ad.6, fd X At what effectve rate of terest s e preset value of a seres of payets of at e ed of every two years, forever, equal to 0? Albert Glover has just sged a cotract w e Blue Jays whch wll pay h 3,000,000 at e begg of each year for e ext fve years. To face hs retreet, e player decdes to put a part of each year s salary (e sae aout each year) to a fud