# SEQUENCES AND SERIES

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1 Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say that a collectio of objects is listed i a sequece, we usually mea that the collectio is ordered i such a way that it has a idetified first member, secod member, third member ad so o. For example, populatio of huma beigs or bacteria at differet times form a sequece. The amout of moey deposited i a bak, over a umber of years form a sequece. Depreciated values of certai commodity occur i a sequece. Sequeces have importat applicatios i several spheres of huma activities. Sequeces, followig specific patters are called progressios. I previous class, we have studied about arithmetic progressio (A.P). I this Chapter, besides discussig more about A.P.; arithmetic mea, geometric mea, relatioship betwee A.M. ad G.M., special series i forms of sum to terms of cosecutive atural umbers, sum to terms of squares of atural umbers ad sum to terms of cubes of atural umbers will also be studied. 9. Sequeces Let us cosider the followig examples: Assume that there is a geeratio gap of 30 years, we are asked to fid the umber of acestors, i.e., parets, gradparets, great gradparets, etc. that a perso might have over 300 years. Here, the total umber of geeratios Fiboacci ( )

2 178 MATHEMATICS The umber of perso s acestors for the first, secod, third,, teth geeratios are, 4, 8, 16, 3,, 104. These umbers form what we call a sequece. Cosider the successive quotiets that we obtai i the divisio of 10 by 3 at differet steps of divisio. I this process we get 3,3.3,3.33,3.333,... ad so o. These quotiets also form a sequece. The various umbers occurrig i a sequece are called its terms. We deote the terms of a sequece by a 1, a, a 3,, a,, etc., the subscripts deote the positio of the term. The th term is the umber at the th positio of the sequece ad is deoted by a. The th term is also called the geeral term of the sequece. Thus, the terms of the sequece of perso s acestors metioed above are: a 1, a 4, a 3 8,, a Similarly, i the example of successive quotiets a 1 3, a 3.3, a ,, a , etc. A sequece cotaiig fiite umber of terms is called a fiite sequece. For example, sequece of acestors is a fiite sequece sice it cotais 10 terms (a fixed umber). A sequece is called ifiite, if it is ot a fiite sequece. For example, the sequece of successive quotiets metioed above is a ifiite sequece, ifiite i the sese that it ever eds. Ofte, it is possible to express the rule, which yields the various terms of a sequece i terms of algebraic formula. Cosider for istace, the sequece of eve atural umbers, 4, 6, Here a 1 1 a 4 a a a , a , ad so o. I fact, we see that the th term of this sequece ca be writte as a, where is a atural umber. Similarly, i the sequece of odd atural umbers 1,3,5,, the th term is give by the formula, a 1, where is a atural umber. I some cases, a arragemet of umbers such as 1, 1,, 3, 5, 8,.. has o visible patter, but the sequece is geerated by the recurrece relatio give by a 1 a 1 a 3 a 1 + a a a + a 1, > This sequece is called Fiboacci sequece.

3 SEQUENCES AND SERIES 179 I the sequece of primes,3,5,7,, we fid that there is o formula for the th prime. Such sequece ca oly be described by verbal descriptio. I every sequece, we should ot expect that its terms will ecessarily be give by a specific formula. However, we expect a theoretical scheme or a rule for geeratig the terms a 1, a, a,,a, i successio. 3 I view of the above, a sequece ca be regarded as a fuctio whose domai is the set of atural umbers or some subset of it of the type {1,, 3...k}. Sometimes, we use the fuctioal otatio a() for a. 9.3 Series Let a 1, a, a,,a, be a give sequece. The, the expressio 3 a 1 + a + a +, + a is called the series associated with the give sequece.the series is fiite or ifiite accordig as the give sequece is fiite or ifiite. Series are ofte represeted i compact form, called sigma otatio, usig the Greek letter (sigma) as meas of idicatig the summatio ivolved. Thus, the series a 1 + a + a a 3 is abbreviated as ak k 1. Remark Whe the series is used, it refers to the idicated sum ot to the sum itself. For example, is a fiite series with four terms. Whe we use the phrase sum of a series, we will mea the umber that results from addig the terms, the sum of the series is 16. We ow cosider some examples. Example 1 Write the first three terms i each of the followig sequeces defied by the followig: (i) a + 5, (ii) a 3. 4 Solutio (i) Here a + 5 Substitutig 1,, 3, we get a 1 (1) + 5 7, a 9, a 3 11 Therefore, the required terms are 7, 9 ad 11. (ii) Here a 3 1. Thus, a 1 3 1,a 1,a

4 180 MATHEMATICS Hece, the first three terms are 1 1, ad 0. 4 Example What is the 0 th term of the sequece defied by a ( 1) ( ) (3 + )? Solutio Puttig 0, we obtai a 0 (0 1) ( 0) (3 + 0) 19 ( 18) (3) Example 3 Let the sequece a be defied as follows: a 1 1, a a 1 + for. Fid first five terms ad write correspodig series. Solutio We have a 1 1, a a , a 3 a , a 4 a , a 5 a Hece, the first five terms of the sequece are 1,3,5,7 ad 9. The correspodig series is EXERCISE 9.1 Write the first five terms of each of the sequeces i Exercises 1 to 6 whose th terms are: 1. a ( + ). a a 4. a a 6 ( 1) a. 4 Fid the idicated terms i each of the sequeces i Exercises 7 to 10 whose th terms are: 7. a 4 3; a 17, a 4 8. a ; a ( ) 9. a ( 1) 1 3 ; a a ; a

5 SEQUENCES AND SERIES 181 Write the first five terms of each of the sequeces i Exercises 11 to 13 ad obtai the correspodig series: 11. a 1 3, a 3a 1 + for all > 1 1. a 1 1, a 13. a 1 a, a a 1 1, > 14. The Fiboacci sequece is defied by 1 a 1 a ad a a 1 + a, >. Fid a+ 1 a, for 1,, 3, 4, 5 a 1, 9.4 Arithmetic Progressio (A.P.) Let us recall some formulae ad properties studied earlier. A sequece a 1, a, a 3,, a, is called arithmetic sequece or arithmetic progressio if a + 1 a + d, N, where a 1 is called the first term ad the costat term d is called the commo differece of the A.P. Let us cosider a A.P. (i its stadard form) with first term a ad commo differece d, i.e., a, a + d, a + d,... The the th term (geeral term) of the A.P. is a a + ( 1) d. We ca verify the followig simple properties of a A.P. : (i) If a costat is added to each term of a A.P., the resultig sequece is also a A.P. (ii) If a costat is subtracted from each term of a A.P., the resultig sequece is also a A.P. (iii) If each term of a A.P. is multiplied by a costat, the the resultig sequece is also a A.P. (iv) If each term of a A.P. is divided by a o-zero costat the the resultig sequece is also a A.P. Here, we shall use the followig otatios for a arithmetic progressio: a the first term, l the last term, d commo differece, the umber of terms. S the sum to terms of A.P. Let a, a + d, a + d,, a + ( 1) d be a A.P. The l a + ( 1) d

6 18 MATHEMATICS S + ( 1) [ a d] a We ca also write, S [ + l ] Let us cosider some examples. Example 4 I a A.P. if m th term is ad the th term is m, where m, fid the pth term. Solutio We have a m a + (m 1) d,... (1) ad a a + ( 1) d m... () Solvig (1) ad (), we get (m ) d m, or d 1,... (3) ad a + m 1... (4) Therefore a p a + (p 1)d + m 1 + ( p 1) ( 1) + m p Hece, the p th term is + m p. 1 Example 5 If the sum of terms of a A.P. is P + ( 1)Q, where P ad Q are costats, fid the commo differece. Solutio Let a 1, a, a be the give A.P. The S a 1 + a + a a 1 + a P + 1 ( 1) Q Therefore S 1 a 1 P, S a 1 + a P + Q So that a S S 1 P + Q Hece, the commo differece is give by d a a 1 (P + Q) P Q. Example 6 The sum of terms of two arithmetic progressios are i the ratio (3 + 8) : (7 + 15). Fid the ratio of their 1 th terms. Solutio Let a 1, a ad d 1, d be the first terms ad commo differece of the first ad secod arithmetic progressio, respectively. Accordig to the give coditio, we have Sum to termsof first A.P Sum to termsof secod A.P

7 SEQUENCES AND SERIES 183 or or Now [ a + ( 1)d ] 1 1 [ a + ( 1)d ] a1+ ( 1) d a + ( 1) d th 1 termof first A.P. a th 1 termof secod A.P a a1+ d a + d d + 11d (1) [By puttig 3 i (1)] th 1+ 11d1 1 term of first A.P. 7 th + 11d 1 term of secod A.P. 16 a Therefore a Hece, the required ratio is 7 : 16. Example 7 The icome of a perso is Rs. 3,00,000, i the first year ad he receives a icrease of Rs.10,000 to his icome per year for the ext 19 years. Fid the total amout, he received i 0 years. Solutio Here, we have a A.P. with a 3,00,000, d 10,000, ad 0. Usig the sum formula, we get, 0 S 0 [ ] 10 (790000) 79,00,000. Hece, the perso received Rs. 79,00,000 as the total amout at the ed of 0 years Arithmetic mea Give two umbers a ad b. We ca isert a umber A betwee them so that a, A, b is a A.P. Such a umber A is called the arithmetic mea (A.M.) of the umbers a ad b. Note that, i this case, we have a+ b A a b A, i.e., A We may also iterpret the A.M. betwee two umbers a ad b as their a+ b average. For example, the A.M. of two umbers 4 ad 16 is 10. We have, thus costructed a A.P. 4, 10, 16 by isertig a umber 10 betwee 4 ad 16. The atural

8 184 MATHEMATICS questio ow arises : Ca we isert two or more umbers betwee give two umbers so that the resultig sequece comes out to be a A.P.? Observe that two umbers 8 ad 1 ca be iserted betwee 4 ad 16 so that the resultig sequece 4, 8, 1, 16 becomes a A.P. More geerally, give ay two umbers a ad b, we ca isert as may umbers as we like betwee them such that the resultig sequece is a A.P. Let A 1, A, A 3,, A be umbers betwee a ad b such that a, A 1, A, A 3,, A, b is a A.P. Here, b is the ( + ) th term, i.e., b a + [( + ) 1]d a + ( + 1) d. b a This gives d. + 1 Thus, umbers betwee a ad b are as follows: A 1 a + d a + b a + 1 A a + d a + ( b a) + 1 3( b a) A 3 a + 3d a A a + d a + b ( a). + 1 Example 8 Isert 6 umbers betwee 3 ad 4 such that the resultig sequece is a A.P. Solutio Let A 1, A, A 3, A 4, A 5 ad A 6 be six umbers betwee 3 ad 4 such that 3, A 1, A, A 3, A 4, A 5, A 6, 4 are i A.P. Here, a 3, b 4, 8. Therefore, (8 1) d, so that d 3. Thus A 1 a + d ; A a + d ; A 3 a + 3d ; A 4 a + 4d ; A 5 a + 5d ; A 6 a + 6d Hece, six umbers betwee 3 ad 4 are 6, 9, 1, 15, 18 ad 1.

10 186 MATHEMATICS 17. A ma starts repayig a loa as first istalmet of Rs If he icreases the istalmet by Rs 5 every moth, what amout he will pay i the 30 th istalmet? 18. The differece betwee ay two cosecutive iterior agles of a polygo is 5. If the smallest agle is 10, fid the umber of the sides of the polygo. 9.5 Geometric Progressio (G. P.) Let us cosider the followig sequeces: (i),4,8,16,..., (ii) 1 1 1,,, 1 (iii). 01,. 0001, , I each of these sequeces, how their terms progress? We ote that each term, except the first progresses i a defiite order. I (i), we have ad so o. 1 a I (ii), we observe, 1 a3 1 a4 1 a1,,, ad so o. 9 a 3 a 3 a Similarly, state how do the terms i (iii) progress? It is observed that i each case, a a a every term except the first term bears a costat a ratio to the term immediately precedig 1 3,,, 4 a1 a a3 1 it. I (i), this costat ratio is ; i (ii), it is ad i (iii), the costat ratio is Such sequeces are called geometric sequece or geometric progressio abbreviated as G.P. A sequece a 1, a, a 3,, a, is called geometric progressio, if each term is ak + 1 o-zero ad r (costat), for k 1. ak By lettig a 1 a, we obtai a geometric progressio, a, ar, ar, ar 3,., where a is called the first term ad r is called the commo ratio of the G.P. Commo ratio i 1 geometric progressio (i), (ii) ad (iii) above are, ad 0.01, respectively. 3 As i case of arithmetic progressio, the problem of fidig the th term or sum of terms of a geometric progressio cotaiig a large umber of terms would be difficult without the use of the formulae which we shall develop i the ext Sectio. We shall use the followig otatios with these formulae: a the first term, r the commo ratio, l the last term,

11 SEQUENCES AND SERIES 187 the umbers of terms, S the sum of terms Geeral term of a G.P. Let us cosider a G.P. with first o-zero term a ad commo ratio r. Write a few terms of it. The secod term is obtaied by multiplyig a by r, thus a ar. Similarly, third term is obtaied by multiplyig a by r. Thus, a 3 a r ar, ad so o. We write below these ad few more terms. 1 st term a 1 a ar 1 1, d term a ar ar 1, 3 rd term a 3 ar ar th term a 4 ar 3 ar 4 1, 5 th term a 5 ar 4 ar 5 1 Do you see a patter? What will be 16 th term? a 16 ar 16 1 ar 15 Therefore, the patter suggests that the th term of a G.P. is give by a ar 1. Thus, a, G.P. ca be writte as a, ar, ar, ar 3, ar 1 ; a, ar, ar,...,ar 1... ;accordig as G.P. is fiite or ifiite, respectively. The series a + ar + ar ar 1 or a + ar + ar ar are called fiite or ifiite geometric series, respectively Sum to terms of a G.P. Let the first term of a G.P. be a ad the commo ratio be r. Let us deote by S the sum to first terms of G.P. The S a + ar + ar ar 1... (1) Case 1 If r 1, we have S a + a + a a ( terms) a Case If r 1, multiplyig (1) by r, we have rs ar + ar + ar ar... () Subtractig () from (1), we get (1 r) S a ar a(1 r ) This gives S a(1 r ) 1 r or S ar ( 1) r 1 Example 9 Fid the 10 th ad th terms of the G.P. 5, 5,15,. Solutio Here a 5 ad r 5. Thus, a 10 5(5) (5) ad a ar 1 5(5) 1 5. Example10 Which term of the G.P.,,8,3,... up to terms is 13107? Solutio Let be the th term of the give G.P. Here a ad r 4. Therefore a (4) 1 or This gives So that 1 8, i.e., 9. Hece, is the 9 th term of the G.P.

12 188 MATHEMATICS Example11 I a G.P., the 3 rd term is 4 ad the 6 th term is 19.Fid the 10 th term. Solutio Here, a3 ar 4... (1) 5 ad a6 ar () Dividig () by (1), we get r. Substitutig r i (1), we get a 6. Hece a 10 6 () Example1 Fid the sum of first terms ad the sum of first 5 terms of the geometric 4 series Solutio Here a 1 ad r 3. Therefore S 1 a(1 r ) 3 1 r I particular, S Example 13 How may terms of the G.P ,,,... are eeded to give the 4 sum ? Solutio Let be the umber of terms eeded. Give that a 3, r 1 ad 3069 S 51 Sice Therefore S a (1 r ) 1 r 1 3(1 )

13 SEQUENCES AND SERIES 189 or or or , which gives 10. Example 14 The sum of first three terms of a G.P. is 13 ad their product is 1. 1 Fid the commo ratio ad the terms. Solutio Let a, a, ar be the first three terms of the G.P. The r a ar a r ad a ( a)( ar ) 1 r From (), we get a 3 1, i.e., a 1 (cosiderig oly real roots) Substitutig a 1 i (1), we have r or 1r r 1 + 5r (1)... () 3 4 This is a quadratic i r, solvig, we get r or Thus, the three terms of G.P. are :, 1, for r ad, 1, for r, Example15 Fid the sum of the sequece 7, 77, 777, 7777,... to terms. Solutio This is ot a G.P., however, we ca relate it to a G.P. by writig the terms as S to terms 7 [ to term] 9 7 [(10 1) (10 1) (10 3 1) ( ) +... terms] 9

18 194 MATHEMATICS 9.7 Sum to Terms of Special Series We shall ow fid the sum of first terms of some special series, amely; (i) (sum of first atural umbers) (ii) (sum of squares of the first atural umbers) (iii) (sum of cubes of the first atural umbers). Let us take them oe by oe. (i) S , the S (ii) Here S ( + 1) (See Sectio 9.4) We cosider the idetity k 3 (k 1) 3 3k 3k + 1 Puttig k 1, successively, we obtai (1) 3 (1) () 3 () (3) 3 (3) ( 1) 3 3 () 3 () + 1 Addig both sides, we get ( ) 3 ( ) k 3 k + k 1 k 1 ( + 1) By (i), we kow that k k 1 3 Hece S 1 3 ( + 1) k + k ( ) 6 ( + 1)(+ 1) 6 (iii) Here S We cosider the idetity, (k + 1) 4 k 4 4k 3 + 6k + 4k + 1 Puttig k 1,, 3, we get

19 SEQUENCES AND SERIES (1) 3 + 6(1) + 4(1) () 3 + 6() + 4() (3) 3 + 6(3) + 4(3) ( 1) 4 ( ) 4 4( ) 3 + 6( ) + 4( ) ( 1) 4 4( 1) 3 + 6( 1) + 4( 1) + 1 ( + 1) Addig both sides, we get ( + 1) ( ) + 6( ) + 4( ) (1) 4 k + 6 k + 4 k + k 1 k 1 k 1 From parts (i) ad (ii), we kow that ( + 1) ( + 1)(+ 1) k ad k 6 k 1 k 1 Puttig these values i equatio (1), we obtai ( + 1) (+ 1) 4 ( + 1) 4 k or k 1 Hece, S 4S ( ) ( + 1) ( + 1). [ ] ( + 1) ( + 1) 4 4 Example 19 Fid the sum to terms of the series: Solutio Let us write S a 1 + a or S a + a 1 + a O subtractio, we get

20 196 MATHEMATICS [ ( 1) terms] a ( 1)[1 + ( ) ] or a ( 1) ( + 4) Hece S k ( 3 1) 3 k 1 k 1 k 1 1 a k + k+ k + k + ( + 1)(+ 1) 3 ( + 1) ( + )( + 4). 3 Example 0 Fid the sum to terms of the series whose th term is (+3). Solutio Give that a ( + 3) + 3 Thus, the sum to terms is give by S k + 3 k 1 k 1 k 1 a k k ( + 1)( + 1) 3 ( + 1) + 6 ( + 1)( + 5). 3 EXERCISE 9.4 Fid the sum to terms of each of the series i Exercises 1 to (1 + ) + ( ) +... Fid the sum to terms of the series i Exercises 8 to 10 whose th terms is give by 8. (+1) (+4) ( 1)

21 SEQUENCES AND SERIES 197 Miscellaeous Examples Example1 If p th, q th, r th ad s th terms of a A.P. are i G.P, the show that (p q), (q r), (r s) are also i G.P. Solutio Here a p a + (p 1) d... (1) a q a + (q 1) d... () a r a + (r 1) d... (3) a s a + (s 1) d... (4) Give that a p, a q, a r ad a s are i G.P., a a a a q r a a a a p q So q r q r p q p q Similarly ar as ar as r s a a a a q r Hece, by (5) ad (6) q r q r (why?) (why?)... (5)... (6) Example If a, b, c are i G.P. ad q r r s, i.e., p q, q r ad r s are i G.P. p q q r x y z a b c, prove that x, y, z are i A.P. Solutio 1 Let x a y b cz k The a k x, b k y ad c k z.... (1) Sice a, b, c are i G.P., therefore, b ac... () Usig (1) i (), we get k y k x + z, which gives y x + z. Hece, x, y ad z are i A.P. Example 3 If a, b, c, d ad p are differet real umbers such that (a + b + c )p (ab + bc + cd) p + (b + c + d ) 0, the show that a, b, c ad d are i G.P. Solutio Give that (a + b + c ) p (ab + bc + cd) p + (b + c + d ) 0... (1)

22 198 MATHEMATICS But L.H.S. (a p abp + b ) + (b p bcp + c ) + (c p cdp + d ), which gives (ap b) + (bp c) + (cp d) 0... () Sice the sum of squares of real umbers is o egative, therefore, from (1) ad (), we have, (ap b) + (bp c) + (cp d) 0 or ap b 0, bp c 0, cp d 0 b c d This implies that p a b c Hece a, b, c ad d are i G.P. Example 4 If p,q,r are i G.P. ad the equatios, px + qx + r 0 ad dx + ex + f 0 have a commo root, the show that d e f,, are i A.P. p q r Solutio The equatio px + qx + r 0 has roots give by q± 4q 4rp x p q Sice p,q, r are i G.P. q pr. Thus x but p dx + ex + f 0 (Why?). Therefore q p is also root of q q d + e + f 0, p p or dq eqp + fp 0... (1) Dividig (1) by pq ad usig q pr, we get d e fp + 0, or e d + f p q pr q p r Hece d e f,, p q r are i A.P.

25 SEQUENCES AND SERIES 01 It took 8 more days to fiish the work. Fid the umber of days i which the work was completed. Summary By a sequece, we mea a arragemet of a umber i a defiite order accordig to some rule. Also, we defie a sequece as a fuctio whose domai is the set of atural umbers or some subsets of the type {1,, 3...k). A sequece cotaiig a fiite umber of terms is called a fiite sequece. A sequece is called ifiite if it is ot a fiite sequece. Let a, a, a,... be the sequece, the the sum expressed as a + a + a is called series. A series is called fiite series if it has got fiite umber of terms. A arithmetic progressio (A.P.) is a sequece i which terms icrease or decrease regularly by the same costat. This costat is called commo differece of the A.P. Usually, we deote the first terms of A.P. by a, the commo differece by d ad the last term by l. The geeral term or the th term of the A.P. is give by a a + ( 1) d. The sum S of the first terms of a A.P. is give by S a+ ( 1) d ( a+l). The arithmetic mea A of ay two umbers a ad b is give by a+b i.e., the sequece a, A, b is i A.P. A sequece is said to be a geometric progressio or G.P., if the ratio of ay term to its precedig term is same throughout. This costat factor is called the commo ratio. Usually, we deote the first term of a G.P. by a ad its commo ratio by r. The geeral or the th term of G.P. is give by a ar 1. The sum S of the first terms of G.P. is give by

26 0 MATHEMATICS ( 1) ( 1 ) a r a r S or,if r 1 r 1 1 r The geometric mea (G.M.) of ay two positive umbers a ad b is give by ab i.e., the sequece a, G, b is G.P. Historical Note Evidece is foud that Babyloias, some 4000 years ago, kew of arithmetic ad geometric sequeces. Accordig to Boethius (510 A.D.), arithmetic ad geometric sequeces were kow to early Greek writers. Amog the Idia mathematicia, Aryabhatta (476 A.D.) was the first to give the formula for the sum of squares ad cubes of atural umbers i his famous work Aryabhatiyam, writte aroud 499 A.D. He also gave the formula for fidig the sum to terms of a arithmetic sequece startig with p th term. Noted Idia mathematicias Brahmgupta (598 A.D.), Mahavira (850 A.D.) ad Bhaskara ( A.D.) also cosidered the sum of squares ad cubes. Aother specific type of sequece havig importat applicatios i mathematics, called Fiboacci sequece, was discovered by Italia mathematicia Leoardo Fiboacci ( A.D.). Seveteeth cetury witessed the classificatio of series ito specific forms. I 1671 A.D. James Gregory used the term ifiite series i coectio with ifiite sequece. It was oly through the rigorous developmet of algebraic ad set theoretic tools that the cocepts related to sequece ad series could be formulated suitably.

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