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 ozero 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.
9 SEQUENCES AND SERIES 185 EXERCISE Fid the sum of odd itegers from 1 to Fid the sum of all atural umbers lyig betwee 100 ad 1000, which are multiples of I a A.P., the first term is ad the sum of the first five terms is oefourth of the ext five terms. Show that 0 th term is How may terms of the A.P. 6, 11, 5, are eeded to give the sum 5? 5. I a A.P., if p th term is 1 q ad qth term is 1, prove that the sum of first pq p terms is 1 (pq +1), where p q. 6. If the sum of a certai umber of terms of the A.P. 5,, 19, is 116. Fid the last term. 7. Fid the sum to terms of the A.P., whose k th term is 5k If the sum of terms of a A.P. is (p + q ), where p ad q are costats, fid the commo differece. 9. The sums of terms of two arithmetic progressios are i the ratio : Fid the ratio of their 18 th terms. 10. If the sum of first p terms of a A.P. is equal to the sum of the first q terms, the fid the sum of the first (p + q) terms. 11. Sum of the first p, q ad r terms of a A.P are. a, b ad c, respectively. a Prove that ( q r) + b ( r p) + c ( p q) 0 p q r 1. The ratio of the sums of m ad terms of a A.P. is m :. Show that the ratio of m th ad th term is (m 1) : ( 1). 13. If the sum of terms of a A.P. is ad its m th term is 164, fid the value of m. 14. Isert five umbers betwee 8 ad 6 such that the resultig sequece is a A.P. a + b 15. If is the A.M. betwee a ad b, the fid the value of. 1 1 a + b 16. Betwee 1 ad 31, m umbers have bee iserted i such a way that the resultig sequece is a A. P. ad the ratio of 7 th ad (m 1) th umbers is 5 : 9. Fid the value of m.
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 ozero 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 ozero 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
14 190 MATHEMATICS 7 [( terms) ( terms)] (10 1) 7 10 (10 1) Example 16 A perso has parets, 4 gradparets, 8 great gradparets, ad so o. Fid the umber of his acestors durig the te geeratios precedig his ow. Solutio Here a, r ad 10 Usig the sum formula S ( a r 1) r 1 We have S 10 ( 10 1) 046 Hece, the umber of acestors precedig the perso is Geometric Mea (G.M.) The geometric mea of two positive umbers a ad b is the umber ab. Therefore, the geometric mea of ad 8 is 4. We observe that the three umbers,4,8 are cosecutive terms of a G.P. This leads to a geeralisatio of the cocept of geometric meas of two umbers. Give ay two positive umbers a ad b, we ca isert as may umbers as we like betwee them to make the resultig sequece i a G.P. Let G 1, G,, G be umbers betwee positive umbers a ad b such that a,g 1,G,G 3,,G,b is a G.P. Thus, b beig the ( + ) th term,we have Hece 1, or b ar + G 1 G 1 1 b + ar a, a b + 1 ar a a b r a G b + 1 ar a a, G b + 1 ar a a, Example17 Isert three umbers betwee 1 ad 56 so that the resultig sequece is a G.P. Solutio Let G 1, G,G 3 be three umbers betwee 1 ad 56 such that 1, G 1,G,G 3,56 is a G.P.
15 SEQUENCES AND SERIES 191 Therefore 56 r 4 givig r ± 4 (Takig real roots oly) For r 4, we have G 1 ar 4, G ar 16, G 3 ar 3 64 Similarly, for r 4, umbers are 4,16 ad 64. Hece, we ca isert, 4, 16, 64 or 4, 16, 64, betwee 1 ad 56 so that the resultig sequeces are i G.P. 9.6 Relatioship Betwee A.M. ad G.M. Let A ad G be A.M. ad G.M. of two give positive real umbers a ad b, respectively. The a+ b A adg ab Thus, we have a+ b a+ b ab A G ab ( a b ) 0... (1) From (1), we obtai the relatioship A G. Example 18 If A.M. ad G.M. of two positive umbers a ad b are 10 ad 8, respectively, fid the umbers. a+ b Solutio Give that A.M (1) ad G.M. ab 8... () From (1) ad (), we get a + b 0... (3) ab (4) Puttig the value of a ad b from (3), (4) i the idetity (a b) (a + b) 4ab, we get (a b) or a b ± 1... (5) Solvig (3) ad (5), we obtai a 4, b 16 or a 16, b 4 Thus, the umbers a ad b are 4, 16 or 16, 4 respectively.
16 19 MATHEMATICS EXERCISE Fid the 0 th ad th terms of the G.P. 5, 5, 5, Fid the 1 th term of a G.P. whose 8 th term is 19 ad the commo ratio is. 3. The 5 th, 8 th ad 11 th terms of a G.P. are p, q ad s, respectively. Show that q ps. 4. The 4 th term of a G.P. is square of its secod term, ad the first term is 3. Determie its 7 th term. 5. Which term of the followig sequeces: (a),, 4,... is 18? (b) 3, 3, 3 3,... is79? (c) ,,,... is? For what values of x, the umbers,x, are i G.P.? 7 7 Fid the sum to idicated umber of terms i each of the geometric progressios i Exercises 7 to 10: , 0.015, ,... 0 terms. 8. 7, 1, 3 7,... terms. 9. 1, a, a, a 3,... terms (if a 1). 10. x 3, x 5, x 7,... terms (if x ± 1). 11. Evaluate 11 k ( + 3 ). k 1 1. The sum of first three terms of a G.P. is 39 ad their product is 1. Fid the 10 commo ratio ad the terms. 13. How may terms of G.P. 3, 3, 3 3, are eeded to give the sum 10? 14. The sum of first three terms of a G.P. is 16 ad the sum of the ext three terms is 18. Determie the first term, the commo ratio ad the sum to terms of the G.P. 15. Give a G.P. with a 79 ad 7 th term 64, determie S Fid a G.P. for which sum of the first two terms is 4 ad the fifth term is 4 times the third term. 17. If the 4 th, 10 th ad 16 th terms of a G.P. are x, y ad z, respectively. Prove that x, y, z are i G.P.
17 SEQUENCES AND SERIES Fid the sum to terms of the sequece, 8, 88, 888, Fid the sum of the products of the correspodig terms of the sequeces, 4, 8, 16, 3 ad 18, 3, 8,, Show that the products of the correspodig terms of the sequeces a, ar, ar, ar 1 ad A, AR, AR, AR 1 form a G.P, ad fid the commo ratio. 1. Fid four umbers formig a geometric progressio i which the third term is greater tha the first term by 9, ad the secod term is greater tha the 4 th by 18.. If the p th, q th ad r th terms of a G.P. are a, b ad c, respectively. Prove that a q r b r p c P q If the first ad the th term of a G.P. are a ad b, respectively, ad if P is the product of terms, prove that P (ab). 4. Show that the ratio of the sum of first terms of a G.P. to the sum of terms from ( + 1) th to () th term is 1. r 5. If a, b, c ad d are i G.P. show that (a + b + c ) (b + c + d ) (ab + bc + cd). 6. Isert two umber betwee 3 ad 81 so that the resultig sequece is G.P. 7. Fid the value of so that a + 1 b may be the geometric mea betwee a + b a ad b. 8. The sum of two umbers is 6 times their geometric meas, show that umbers are i the ratio ( 3+ ) :( 3 ). 9. If A ad G be A.M. ad G.M., respectively betwee two positive umbers, prove that the umbers are A ± ( A+ G)( A G). 30. The umber of bacteria i a certai culture doubles every hour. If there were 30 bacteria preset i the culture origially, how may bacteria will be preset at the ed of d hour, 4 th hour ad th hour? 31. What will Rs 500 amouts to i 10 years after its deposit i a bak which pays aual iterest rate of 10% compouded aually? 3. If A.M. ad G.M. of roots of a quadratic equatio are 8 ad 5, respectively, the obtai the quadratic equatio.
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.
23 SEQUENCES AND SERIES 199 Miscellaeous Exercise O Chapter 9 1. Show that the sum of (m + ) th ad (m ) th terms of a A.P. is equal to twice the m th term.. If the sum of three umbers i A.P., is 4 ad their product is 440, fid the umbers. 3. Let the sum of,, 3 terms of a A.P. be S 1, S ad S 3, respectively, show that S 3 3(S S 1 ) 4. Fid the sum of all umbers betwee 00 ad 400 which are divisible by Fid the sum of itegers from 1 to 100 that are divisible by or Fid the sum of all two digit umbers which whe divided by 4, yields 1 as remaider. 7. If f is a fuctio satisfyig f (x +y) f(x) f(y) for all x, y N such that f(1) 3 ad f( x) 10, fid the value of. x 1 8. The sum of some terms of G.P. is 315 whose first term ad the commo ratio are 5 ad, respectively. Fid the last term ad the umber of terms. 9. The first term of a G.P. is 1. The sum of the third term ad fifth term is 90. Fid the commo ratio of G.P. 10. The sum of three umbers i G.P. is 56. If we subtract 1, 7, 1 from these umbers i that order, we obtai a arithmetic progressio. Fid the umbers. 11. A G.P. cosists of a eve umber of terms. If the sum of all the terms is 5 times the sum of terms occupyig odd places, the fid its commo ratio. 1. The sum of the first four terms of a A.P. is 56. The sum of the last four terms is 11. If its first term is 11, the fid the umber of terms. 13. If a + bx b cx c dx + + ( a bx b cx c dx x 0 ), the show that a, b, c ad d are i G.P. 14. Let S be the sum, P the product ad R the sum of reciprocals of terms i a G.P. Prove that P R S. 15. The p th, q th ad r th terms of a A.P. are a, b, c, respectively. Show that (q r )a + (r p )b + (p q )c If a +,b +,c + are i A.P., prove that a, b, c are i A.P. b c c a a b 17. If a, b, c, d are i G.P, prove that (a + b ), (b + c ), (c + d ) are i G.P. 18. If a ad b are the roots of x 3x + p 0 ad c, d are roots of x 1x + q 0, where a, b, c, d form a G.P. Prove that (q + p) : (q p) 17:15.
24 00 MATHEMATICS 19. The ratio of the A.M. ad G.M. of two positive umbers a ad b, is m :. Show that a: b ( m m ) ( ) : m m If a, b, c are i A.P.; b, c, d are i G.P. ad 1, 1, 1 are i A.P. prove that a, c, e c d e are i G.P. 1. Fid the sum of the followig series up to terms: (i) (ii) Fid the 0 th term of the series terms. 3. Fid the sum of the first terms of the series: If S 1, S, S 3 are the sum of first atural umbers, their squares ad their cubes, respectively, show that 9S S 3 (1 + 8S 1 ). 5. Fid the sum of the followig series up to terms: ( + 1) Show that ( + 1) A farmer buys a used tractor for Rs He pays Rs 6000 cash ad agrees to pay the balace i aual istalmets of Rs 500 plus 1% iterest o the upaid amout. How much will the tractor cost him? 8. Shamshad Ali buys a scooter for Rs 000. He pays Rs 4000 cash ad agrees to pay the balace i aual istalmet of Rs 1000 plus 10% iterest o the upaid amout. How much will the scooter cost him? 9. A perso writes a letter to four of his frieds. He asks each oe of them to copy the letter ad mail to four differet persos with istructio that they move the chai similarly. Assumig that the chai is ot broke ad that it costs 50 paise to mail oe letter. Fid the amout spet o the postage whe 8 th set of letter is mailed. 30. A ma deposited Rs i a bak at the rate of 5% simple iterest aually. Fid the amout i 15 th year sice he deposited the amout ad also calculate the total amout after 0 years. 31. A maufacturer reckos that the value of a machie, which costs him Rs. 1565, will depreciate each year by 0%. Fid the estimated value at the ed of 5 years workers were egaged to fiish a job i a certai umber of days. 4 workers dropped out o secod day, 4 more workers dropped out o third day ad so o.
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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