# 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 real-life problems. Why you should lear it Arithmetic sequeces have practical real-life 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

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.

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 oe-half brick wide ad the bottom row is 14 bricks wide. How may bricks are used i the fiished wall?

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