Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...


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1 3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces. Fid th partial sums of arithmetic sequeces. Use arithmetic sequeces to model ad solve reallife problems. Why you should lear it Arithmetic sequeces have practical reallife applicatios. For istace, i Exercise 83 o page 660, a arithmetic sequece is used to model the seatig capacity of a auditorium. Arithmetic Sequeces A sequece whose cosecutive terms have a commo differece is called a arithmetic sequece. Defiitio of Arithmetic Sequece A sequece is arithmetic if the differeces betwee cosecutive terms are the same. So, the sequece a 1, a, a 3, a 4,...,,... is arithmetic if there is umber d such that a a 1 a 3 a a 4 a 3... d. The umber d is the commo differece of the arithmetic sequece. Example 1 Examples of Arithmetic Sequeces a. The sequece whose th term is 4 3 is arithmetic. For this sequece, the commo differece betwee cosecutive terms is 4. 7, 11, 15, 19,..., 4 3,... Begi with mediacolor s Alamy b. The sequece whose th term is 7 5 is arithmetic. For this sequece, the commo differece betwee cosecutive terms is 5., 3, 8, 13,..., 7 5,... Begi with c. The sequece whose th term is 4 3 is arithmetic. For this sequece, the commo differece betwee cosecutive terms is 1, 5 4, 3, 7 4,..., 3 4, Begi with Now try Exercise 1. The sequece 1, 4, 9, 16,..., whose th term is, is ot arithmetic. The differece betwee the first two terms is a a but the differece betwee the secod ad third terms is a 3 a
2 3330_090.qxd 1/5/05 11:9 AM Page Chapter 9 Sequeces, Series, ad Probability I Example 1, otice that each of the arithmetic sequeces has a th term that is of the form d c, where the commo differece of the sequece is d. A arithmetic sequece may be thought of as a liear fuctio whose domai is the set of atural umbers. = d + c c a 1 a a 3 FIGURE 9.3 The th Term of a Arithmetic Sequece The th term of a arithmetic sequece has the form d c Liear form where d is the commo differece betwee cosecutive terms of the sequece ad c a 1 d. A graphical represetatio of this defiitio is show i Figure 9.3. Substitutig a 1 d for c i d c yields a alterative recursio form for the th term of a arithmetic sequece. a 1 1 d Alterative form Example Fidig the th Term of a Arithmetic Sequece The alterative recursio form of the th term of a arithmetic sequece ca be derived from the patter below. a 1 a 1 a a 1 d a 3 a 1 d a 4 a 1 3d a 5 a 1 4d 1 less a 1 1 d 1 less 1st term d term 3rd term 4th term 5th term th term As a aid to learig the formula for the th term of a arithmetic sequece, cosider havig your studets ituitively fid the th term of each of the followig sequeces. 1. 5, 8, 11, 14, 17,... Aswer: 3. a, a, a 4, a 6,... Aswer: a Fid a formula for the th term of the arithmetic sequece whose commo differece is 3 ad whose first term is. Solutio Because the sequece is arithmetic, you kow that the formula for the th term is of the form d c. Moreover, because the commo differece is d 3, the formula must have the form 3 c. Because a 1, it follows that c a 1 d So, the formula for the th term is 3 1. Substitute 3 for d. Substitute for The sequece therefore has the followig form., 5, 8, 11, 14,..., 3 1,... Now try Exercise 1. ad 3 for d. Aother way to fid a formula for the th term of the sequece i Example is to begi by writig the terms of the sequece. a a 3 5 a From these terms, you ca reaso that the th term is of the form d c 3 1. a a 1 a a a
3 3330_090.qxd 1/5/05 11:9 AM Page 655 Sectio 9. Arithmetic Sequeces ad Partial Sums 655 Example 3 Writig the Terms of a Arithmetic Sequece You ca fid a 1 i Example 3 by usig the alterative recursio form of the th term of a arithmetic sequece, as follows. a 1 1d a 4 a 1 4 1d 0 a a a 1 The fourth term of a arithmetic sequece is 0, ad the 13th term is 65. Write the first 11 terms of this sequece. Solutio You kow that a 4 0 ad a So, you must add the commo differece d ie times to the fourth term to obtai the 13th term. Therefore, the fourth ad 13th terms of the sequece are related by a 13 a 4 9d. ad are ie terms apart. Usig a 4 0 ad a 13 65, you ca coclude that d 5, which implies that the sequece is as follows. a 1 5 a 10 a 3 15 a 4 0 a 4 a 5 5 a 13 a 6 30 Now try Exercise 37. a 7 35 a 8 40 a 9 45 a a If you kow the th term of a arithmetic sequece ad you kow the commo differece of the sequece, you ca fid the 1th term by usig the recursio formula 1 d. Recursio formula With this formula, you ca fid ay term of a arithmetic sequece, provided that you kow the precedig term. For istace, if you kow the first term, you ca fid the secod term. The, kowig the secod term, you ca fid the third term, ad so o. Example 4 Usig a Recursio Formula Fid the ith term of the arithmetic sequece that begis with ad 9. Solutio For this sequece, the commo differece is d 9 7. There are two ways to fid the ith term. Oe way is simply to write out the first ie terms (by repeatedly addig 7)., 9, 16, 3, 30, 37, 44, 51, 58 Aother way to fid the ith term is to first fid a formula for the th term. Because the first term is, it follows that c a 1 d 7 5. Therefore, a formula for the th term is 7 5 which implies that the ith term is a Now try Exercise 45.
4 3330_090.qxd 1/5/05 11:9 AM Page Chapter 9 Sequeces, Series, ad Probability The Sum of a Fiite Arithmetic Sequece There is a simple formula for the sum of a fiite arithmetic sequece. Note that this formula works oly for arithmetic sequeces. The Sum of a Fiite Arithmetic Sequece The sum of a fiite arithmetic sequece with terms is S a 1. For a proof of the sum of a fiite arithmetic sequece, see Proofs i Mathematics o page 73. Example 5 Fidig the Sum of a Fiite Arithmetic Sequece Fid the sum: Solutio To begi, otice that the sequece is arithmetic (with a commo differece of ). Moreover, the sequece has 10 terms. So, the sum of the sequece is S a 1 Formula for the sum of a arithmetic sequece Substitute 10 for, 1 for a 1, ad 19 for. 50. Now try Exercise 63. Historical Note A teacher of Carl Friedrich Gauss ( ) asked him to add all the itegers from 1 to. Whe Gauss retured with the correct aswer after oly a few momets, the teacher could oly look at him i astouded silece. This is what Gauss did: S S S S The Grager Collectio Fidig the Sum of a Fiite Arithmetic Sequece Fid the sum of the itegers (a) from 1 to ad (b) from 1 to N. Solutio a. The itegers from 1 to form a arithmetic sequece that has terms. So, you ca use the formula for the sum of a arithmetic sequece, as follows. b. Example 6 S a a 1 N 1 N Formula for sum of a arithmetic sequece Substitute for, 1 for a 1, for. S N Now try Exercise 65. Formula for sum of a arithmetic sequece Substitute N for, 1 for a 1, ad N for.
5 3330_090.qxd 1/5/05 11:9 AM Page 657 Sectio 9. Arithmetic Sequeces ad Partial Sums 657 The sum of the first terms of a ifiite sequece is the th partial sum. The th partial sum ca be foud by usig the formula for the sum of a fiite arithmetic sequece. Example 7 Fidig a Partial Sum of a Arithmetic Sequece Fid the 150th partial sum of the arithmetic sequece 5, 16, 7, 38, 49,.... Solutio For this arithmetic sequece, a 1 5 ad d So, c a 1 d ad the th term is Therefore, a , ad the sum of the first 150 terms is S 150 a 1 a 150 th partial sum formula ,675. Applicatios Now try Exercise 69. Substitute 150 for, 5 for a 1, ad 1644 for th partial sum a 150. Example 8 Prize Moey I a golf touramet, the 16 golfers with the lowest scores wi cash prizes. First place receives a cash prize of $0, secod place receives $950, third place receives $900, ad so o. What is the total amout of prize moey? Solutio The cash prizes awarded form a arithmetic sequece i which the commo differece is d 50. Because c a 1 d you ca determie that the formula for the th term of the sequece is So, the 16th term of the sequece is a , ad the total amout of prize moey is S S 16 a 1 a 16 th partial sum formula Substitute 16 for, 0 for a 1, ad 50 for a $10,000. Now try Exercise 89.
6 3330_090.qxd 1/5/05 11:30 AM Page Chapter 9 Sequeces, Series, ad Probability Example 9 Total Sales Activities 1. Determie which of the followig are arithmetic sequeces. (a) 3, 5, 7, 9, 11,... (b) 3, 6, 1, 4, 48,... (c) 3, 6, 9, 1, 15,... (d) 5, 0, 5, 10, 15,... (e) 1, 3, 6, 10, 15, 1,... Aswer: (a) ad (d). Fid the first five terms of the arithmetic sequece with a 1 13 ad d 4. Aswer: 13, 9, 5, 1, 3 3. Fid the sum. Sales (i dollars) 3 1 Aswer: 15,350 80,000 60,000 40,000 0,000 FIGURE 9.4 Small Busiess = Year A small busiess sells $10,000 worth of ski care products durig its first year. The ower of the busiess has set a goal of icreasig aual sales by $7500 each year for 9 years. Assumig that this goal is met, fid the total sales durig the first 10 years this busiess is i operatio. Solutio The aual sales form a arithmetic sequece i which a 1 10,000 d So, c a 1 d 10, ad the th term of the sequece is This implies that the 10th term of the sequece is a ,500. See Figure 9.4. The sum of the first 10 terms of the sequece is S 10 a 1 a 10 th partial sum formula 10 10,000 77,500 Substitute 10 for, 10,000 for a 1, ad 77,500 for a , ,500. So, the total sales for the first 10 years will be $437,500. Now try Exercise 91. ad W RITING ABOUT MATHEMATICS Numerical Relatioships Decide whether it is possible to fill i the blaks i each of the sequeces such that the resultig sequece is arithmetic. If so, fid a recursio formula for the sequece. a. 7,,,,,,11 b. 17,,,,,,,,,71 c., 6,,, 16 d. 4, 7.5,,,,,,,,,39 e. 8, 1,,,, 60.75
7 3330_090.qxd 1/5/05 11:30 AM Page 659 Sectio 9. Arithmetic Sequeces ad Partial Sums Exercises VOCABULARY CHECK: Fill i the blaks. 1. A sequece is called a sequece if the differeces betwee two cosecutive terms are the same. This differece is called the differece.. The th term of a arithmetic sequece has the form. 3. The formula S ca be used to fid the sum of the first terms of a arithmetic sequece, a 1 called the of a. PREREQUISITE SKILLS REVIEW: Practice ad review algebra skills eeded for this sectio at I Exercises 1 10, determie whether the sequece is arithmetic. If so, fid the commo differece , 8, 6, 4,,.... 4, 7, 10, 13, 16, ,, 4, 8, 16, , 40, 0, 10, 5, ,,, 3 4,, 7 4, 3, 5 4,..., 1, , 3, 1, 3, 6, , 5.7, 6.1, 6.5, 6.9, l 1, l, l 3, l 4, l 5, ,, 3, 4, 5,... I Exercises 11 18, write the first five terms of the sequece. Determie whether the sequece is arithmetic. If so, fid the commo differece. (Assume that begis with 1.) I Exercises 19 30, fid a formula for sequece. 19. a 1 1, d 3 0. a 1 15, d 4 1. a 1, d 8. a 1 0, d 3 3. a 1 x, d x 4. a 1 y, d 5y 5. 4, 3, 1, 7, , 5, 0, 5, 10, a 1 5, a 4 15 for the arithmetic 8. a 1 4, a a 3 94, a a 5 190, a I Exercises 31 38, write the first five terms of the arithmetic sequece. 31. a 1 5, d 6 3. a 1 5, d a 1.6, d a , d a 1, a a 4 16, a a 8 6, a a 3 19, a I Exercises 39 44, write the first five terms of the arithmetic sequece. Fid the commo differece ad write the th term of the sequece as a fuctio of. 39. a 1 15, a k1 a k a 1 6, a k1 a k a 1 00, a k1 a k a 1 7, a k1 a k a 1 5 8, a k1 a k a , a k1 a k 0.5 I Exercises 45 48, the first two terms of the arithmetic sequece are give. Fid the missig term. 45. a 1 5, a 11, a a 1 3, a 13, a a 1 4., a 6.6, a a 1 0.7, a 13.8, a 8
8 3330_090.qxd 1/5/05 11:30 AM Page Chapter 9 Sequeces, Series, ad Probability I Exercises 49 5, match the arithmetic sequece with its graph. [The graphs are labeled (a), (b), (c), ad (d).] (a) (c) (b) (d) I Exercises 53 56, use a graphig utility to graph the first 10 terms of the sequece. (Assume that begis with 1.) I Exercises 57 64, fid the idicated th partial sum of the arithmetic sequece , 0, 3, 44,..., 58., 8, 14, 0,..., , 3.7, 3.,.7,..., , 0.9, 1.3, 1.7,..., , 37, 34, 31,..., 6. 75, 70, 65, 60,..., a 1, a 5 0, a 1 15, a 307, Fid the sum of the first positive odd itegers. 66. Fid the sum of the itegers from 10 to 50. I Exercises 67 74, fid the partial sum I Exercises 75 80, use a graphig utility to fid the partial sum i i Job Offer I Exercises 81 ad 8, cosider a job offer with the give startig salary ad the give aual raise. (a) Determie the salary durig the sixth year of employmet. (b) Determie the total compesatio from the compay through six full years of employmet. Startig Salary Aual Raise 81. $3,500 $ $36,800 $ Seatig Capacity Determie the seatig capacity of a auditorium with 30 rows of seats if there are 0 seats i the first row, 4 seats i the secod row, 8 seats i the third row, ad so o. 84. Seatig Capacity Determie the seatig capacity of a auditorium with 36 rows of seats if there are 15 seats i the first row, 18 seats i the secod row, 1 seats i the third row, ad so o. 85. Brick Patter A brick patio has the approximate shape of a trapezoid (see figure). The patio has 18 rows of bricks. The first row has 14 bricks ad the 18th row has 31 bricks. How may bricks are i the patio? j j1 FIGURE FOR 85 FIGURE FOR Brick Patter A triagular brick wall is made by cuttig some bricks i half to use i the first colum of every other row. The wall has 8 rows. The top row is oehalf brick wide ad the bottom row is 14 bricks wide. How may bricks are used i the fiished wall?
9 3330_090.qxd 1/5/05 11:30 AM Page 661 Sectio 9. Arithmetic Sequeces ad Partial Sums Fallig Object A object with egligible air resistace is dropped from a plae. Durig the first secod of fall, the object falls 4.9 meters; durig the secod secod, it falls 14.7 meters; durig the third secod, it falls 4.5 meters; durig the fourth secod, it falls 34.3 meters. If this arithmetic patter cotiues, how may meters will the object fall i 10 secods? 88. Fallig Object A object with egligible air resistace is dropped from the top of the Sears Tower i Chicago at a height of 1454 feet. Durig the first secod of fall, the object falls 16 feet; durig the secod secod, it falls 48 feet; durig the third secod, it falls 80 feet; durig the fourth secod, it falls 11 feet. If this arithmetic patter cotiues, how may feet will the object fall i 7 secods? 89. Prize Moey A couty fair is holdig a baked goods competitio i which the top eight bakers receive cash prizes. First places receives a cash prize of $00, secod place receives $175, third place receives $150, ad so o. (a) Write a sequece that represets the cash prize awarded i terms of the place i which the baked good places. (b) Fid the total amout of prize moey awarded at the competitio. 90. Prize Moey A city bowlig league is holdig a touramet i which the top 1 bowlers with the highest threegame totals are awarded cash prizes. First place will wi $, secod place $1, third place $0, ad so o. (a) Write a sequece that represets the cash prize awarded i terms of the place i which the bowler fiishes. (b) Fid the total amout of prize moey awarded at the touramet. 91. Total Profit A small sowplowig compay makes a profit of $8000 durig its first year. The ower of the compay sets a goal of icreasig profit by $1500 each year for 5 years. Assumig that this goal is met, fid the total profit durig the first 6 years of this busiess. What kids of ecoomic factors could prevet the compay from meetig its profit goal? Are there ay other factors that could prevet the compay from meetig its goal? Explai. 9. Total Sales A etrepreeur sells $15,000 worth of sports memorabilia durig oe year ad sets a goal of icreasig aual sales by $5000 each year for 9 years. Assumig that this goal is met, fid the total sales durig the first 10 years of this busiess. What kids of ecoomic factors could prevet the busiess from meetig its goals? 93. Borrowig Moey You borrowed $000 from a fried to purchase ew laptop computer ad have agreed to pay back the loa with mothly paymets of $00 plus 1% iterest o the upaid balace. (a) Fid the first six mothly paymets you will make, ad the upaid balace after each moth. (b) Fid the total amout of iterest paid over the term of the loa. 94. Borrowig Moey You borrowed $5000 from your parets to purchase a used car. The arragemets of the loa are such that you will make paymets of $50 per moth plus 1% iterest o the upaid balace. (a) Fid the first year s mothly paymets you will make, ad the upaid balace after each moth. (b) Fid the total amout of iterest paid over the term of the loa. Model It 95. Data Aalysis: Persoal Icome The table shows the per capita persoal icome i the Uited States from 1993 to 003. (Source: U.S. Bureau of Ecoomic Aalysis) Year Per capita persoal icome, 1993 $1, $, $3, $4, $5, $6, $7, $9, $30, $30, $31,633 (a) Fid a arithmetic sequece that models the data. Let represet the year, with 3 correspodig to (b) Use the regressio feature of a graphig utility to fid a liear model for the data. How does this model compare with the arithmetic sequece you foud i part (a)? (c) Use a graphig utility to graph the terms of the fiite sequece you foud i part (a). (d) Use the sequece from part (a) to estimate the per capita persoal icome i 004 ad 005. (e) Use your school s library, the Iteret, or some other referece source to fid the actual per capita persoal icome i 004 ad 005, ad compare these values with the estimates from part (d).
10 3330_090.qxd 1/8/05 10:53 AM Page Chapter 9 Sequeces, Series, ad Probability 96. Data Aalysis: Reveue The table shows the aual reveue (i millios of dollars) for Nextel Commuicatios, Ic. from 1997 to 003. (Source: Nextel Commuicatios, Ic.) (a) Costruct a bar graph showig the aual reveue from 1997 to 003. (b) Use the liear regressio feature of a graphig utility to fid a arithmetic sequece that approximates the aual reveue from 1997 to 003. (c) Use summatio otatio to represet the total reveue from 1997 to 003. Fid the total reveue. (d) Use the sequece from part (b) to estimate the aual reveue i 008. Sythesis Year Reveue, ,80 True or False? I Exercises 97 ad 98, determie whether the statemet is true or false. Justify your aswer. 97. Give a arithmetic sequece for which oly the first two terms are kow, it is possible to fid the th term. 98. If the oly kow iformatio about a fiite arithmetic sequece is its first term ad its last term, the it is possible to fid the sum of the sequece. 99. Writig I your ow words, explai what makes a sequece arithmetic.. Writig Explai how to use the first two terms of a arithmetic sequece to fid the th term Exploratio (a) Graph the first 10 terms of the arithmetic sequece 3. (b) Graph the equatio of the lie y 3x. (c) Discuss ay differeces betwee the graph of 3 ad the graph of y 3x. (d) Compare the slope of the lie i part (b) with the commo differece of the sequece i part (a). What ca you coclude about the slope of a lie ad the commo differece of a arithmetic sequece? 10. Patter Recogitio (a) Compute the followig sums of positive odd itegers. (b) Use the sums i part (a) to make a cojecture about the sums of positive odd itegers. Check your cojecture for the sum (c) Verify your cojecture algebraically Thik About It The sum of the first 0 terms of a arithmetic sequece with a commo differece of 3 is 650. Fid the first term Thik About It The sum of the first terms of a arithmetic sequece with first term a 1 ad commo differece d is S. Determie the sum if each term is icreased by 5. Explai. Skills Review I Exercises , fid the slope ad yitercept (if possible) of the equatio of the lie. Sketch the lie x 4y x y x y 11 0 I Exercises 109 ad 110, use GaussJorda elimiatio to solve the system of equatios x 3x 6x x 5x 8x y 7z 10 y 4z 17 5y z 0 4y 3y y 10z z 3z Make a Decisio To work a exteded applicatio aalyzig the media sales price of existig oefamily homes i the Uited States from 1987 to 003, visit this text s website at college.hmco.com. (Data Source: Natioal Associatio of Realtors)
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