SEQUENCE AND SERIES. Chapter Overview

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1 Chapter 9 SEQUENCE AND SERIES 9.1 Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a 1, a,..., etc., the subscript deotes the positio of the term. I view of the above a sequece i the set X ca be regarded as a mappig or a fuctio f : N X defied by f () t N. Domai of f is a set of atural umbers or some subset of it deotig the positio of term. If its rage deotig the value of terms is a subset of R real umbers the it is called a real sequece. A sequece is either fiite or ifiite depedig upo the umber of terms i a sequece. We should ot expect that its terms will be ecessarily give by a specific formula. However, we expect a theoretical scheme or rule for geeratig the terms. Let a 1, a,..., be the sequece, the, the expressio a 1 + a + a is called the series associated with give sequece. The series is fiite or ifiite accordig as the give sequece is fiite or ifiite. Remark Whe the series is used, it refers to the idicated sum ot to the sum itself. Sequece followig certai patters are more ofte called progressios. I progressios, we ote that each term except the first progresses i a defiite maer Arithmetic progressio (A.P.) is a sequece i which each term except the first is obtaied by addig a fixed umber (positive or egative) to the precedig term. Thus ay sequece a 1, a... a,... is called a arithmetic progressio if a + 1 a + d, N, where d is called the commo differece of the A.P., usually we deote the first term of a A.P by a 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 th term from the last is give by a l ( 1) d

2 148 EXEMPLAR PROBLEMS MATHEMATICS The sum S of the first terms of a A.P. is give by S a ( 1) d ( a l), where l a + ( 1) d is the last terms of the A.P., ad the geeral term is give by a S S 1 The arithmetic mea for ay positive umbers a 1, a,... a is give by a1 a... a A.M. If a, A ad b are i A.P., the A is called the arithmetic mea of umbers a ad b ad i.e., A a b If the terms of a A.P. are icreased, decreased, multiplied or divided by the same costat, they still remai i A.P. If a 1, a... are i A.P. with commo differece d, the (i) a 1 ± k, a ± k ± k,... are also i A.P with commo differece d. (ii) a 1 k, a k k,... are also i A.P with commo differece dk (k 0). 1 3 ad,, a k k k... are also i A.P. with commo differece d (k 0). k If a 1, a... ad b 1, b, b 3... are two A.P., the (i) a 1 ± b 1, a ± b ± b 3,... are also i A.P (ii) a 1 3 a 1 b 1, a b b 3,... ad,, a b1 b b,... are ot i A.P. 3 If a 1, a... ad a are i A.Ps, the (i) a 1 + a a + a 1 a 3 + a... (ii) ar k ar k ar k, 0 k r (iii) (iv) If th term of ay sequece is liear expressio i, the the sequece is a A.P. If sum of terms of ay sequece is a quadratic expressio i, the sequece is a A.P.

3 SEQUENCE AND SERIES A Geometric progressio (G.P.) is a sequece i which each term except the first is obtaied by multiplyig the previous term by a o-zero costat called the commo ratio. Let us cosider a G.P. with first o-zero term a ad commo ratio r, i.e., a, ar, ar,..., ar 1,... 1 ar Here, commo ratio r ar The geeral term or th term of G.P. is give by a ar 1. Last term l of a G.P. is same as the th term ad is give by l ar 1. l ad the th term from the last is give by a 1 The sum S of the first terms is give by r S a( r 1), if r 1 r 1 S a if r 1 If a, G ad b are i G.P., the G is called the geometric mea of the umbers a ad b ad is give by (i) (ii) G If the terms of a G.P. are multiplied or divided by the same o-zero costat (k 0), they still remai i G.P. ab If a 1, a,..., are i G.P., the a 1 k, a k k,... ad are also i G.P. with same commo ratio, i particularly if a 1, a,... are i G.P., the 1 1 1,, a a a,... are also i G.P. 1 3 a 1 3, a, a k k k,... If a 1, a,... ad b 1, b, b 3,... are two G.P.s, the a 1 b 1, a b b 3,... ad a 1 3, a, a b b b,... are also i G.P. 1 3 (iii) If a 1, a,... are i A.P. (a i > 0 i), the a1 a a3 x, x, x,..., are i G.P. ( x > 0)

4 150 EXEMPLAR PROBLEMS MATHEMATICS (iv) If a 1, a,..., a are i G.P., the a 1 a a a 1 a 3 a Importat results o the sum of special sequeces (i) Sum of the first atural umbers: (ii) ( 1) Sum of the squares of first atural umbers. ( 1)(1) (iii) Sum of cubes of first atural umbers: ( 1) Solved Examples Short Aswer Type Example 1 The first term of a A.P. is a, the secod term is b ad the last term is c. Show that the sum of the A.P. is ( b c a )( c a ). ( b a) Solutio Let d be the commo diffrece ad be the umber of terms of the A.P. Sice the first term is a ad the secod term is b Therefore, d b a Also, the last term is c, so c a + ( 1) (b a) (sice d b a) 1 c a b a 1 + c a b a b a c a bc a ba b a ( b c a) Therefore, S ( a l ) ( a c ) ( b a) Example The p th term of a A.P. is a ad q th term is b. Prove that the sum of its (p + q) terms is

5 SEQUENCE AND SERIES 151 p q ab a b p q. Solutio Let A be the first term ad D be the commo differece of the A.P. It is give that t p a A + (p 1) D a... (1) t q b A + ( q 1) D b... () Subtractig () from (1), we get (p 1 q + 1) D a b D a b p q Addig (1) ad (), we get A + (p + q ) D a + b A + (p + q 1) D a + b + D a b A + (p + q 1) D a + b + p q... (3).. (4) Now S p + q p q [A + (p + q 1) D] p q a b a b p q [(usig... (3) ad (4)] Example 3 If there are ( + 1) terms i a A.P., the prove that the ratio of the sum of odd terms ad the sum of eve terms is ( + 1) : Solutio Let a be the first term ad d the commo differece of the A.P. Also let S 1 be the sum of odd terms of A.P. havig ( + 1) terms. The S 1 a 1 + a 3 + a a S 1 ( a1 a 1 ) 1 S 1 a a (1 1) d

6 15 EXEMPLAR PROBLEMS MATHEMATICS ( + 1) (a + d) Similarly, if S deotes the sum of eve terms, the S [a + d] (a + d) Hece S1 S ( 1)( ad ) 1 ( a d) Example 4 At the ed of each year the value of a certai machie has depreciated by 0% of its value at the begiig of that year. If its iitial value was Rs 150, fid the value at the ed of 5 years. Solutio After each year the value of the machie is 80% of its value the previous year so at the ed of 5 years the machie will depreciate as may times as 5. Hece, we have to fid the 6 th term of the G.P. whose first term a 1 is 150 ad commo ratio r is.8. Hece, value at the ed 5 years t 6 a 1 r (.8) Example 5 Fid the sum of first 4 terms of the A.P. a 1, a,... if it is kow that a 1 + a 5 + a 10 + a 15 + a 0 + a 4 5. Solutio We kow that i a A.P., the sum of the terms equidistat from the begiig ad ed is always the same ad is equal to the sum of first ad last term. Therefore d b a i.e., a 1 + a 4 a 5 + a 0 a 10 + a 15 It is give that (a 1 + a 4 ) + (a 5 + a 0 ) + (a 10 + a 15 ) 5 (a 1 + a 4 ) + (a 1 + a 4 ) + (a 1 + a 4 ) 5 3 (a 1 + a 4 ) 5 a 1 + a 4 75 We kow that S [ a l ], where a is the first term ad l is the last term of a A.P. Thus, S 4 4 [a + a 1 4 ] Example 6 The product of three umbers i A.P. is 4, ad the largest umber is 7 times the smallest. Fid the umbers. Solutio Let the three umbers i A.P. be a d, a, a + d (d > 0)

7 SEQUENCE AND SERIES 153 Now (a d) a (a + d) 4 a (a d ) 4... (1) Now, sice the largest umber is 7 times the smallest, i.e., a + d 7 (a d) Therefore, d 3 a 4 Substitutig this value of d i (1), we get 9a a a 16 4 a 8 ad d 3 a Hece, the three umbers are, 8, 14. Example 7 Show that (x + xy + y ), (z + xz + x ) ad (y + yz + z ) are cosecutive terms of a A.P., if x, y ad z are i A.P. Solutio The terms (x + xy + y ), (z + xz + x ) ad (y + yz + z ) will be i A.P. if (z + xz + x ) (x + xy + y ) (y + yz + z ) (z + xz + x ) i.e., z + xz xy y y + yz xz x i.e., x + z + xz y y + yz + xy i.e., (x + z) y y (x + y + z) i.e., x + z y y i.e., x + z y which is true, sice x, y, z are i A.P. Hece x + xy + y, z + xz + x, y + yz + z are i A.P. Example 8 If a, b, c, d are i G.P., prove that a b, b c, c d are also i G.P. Solutio Let r be the commo ratio of the give G.P. The b c d r a b c b ar, c br ar, d cr ar 3 Now, a b a a r a (1 r )

8 154 EXEMPLAR PROBLEMS MATHEMATICS b c a r a r 4 a r (1 r ) ad c d a r 4 a r 6 a r 4 (1 r ) Therefore, b a c b Hece, a b, b c, c d are i G.P. c b d c r Log Aswer Type Example 9 If the sum of m terms of a A.P. is equal to the sum of either the ext terms or the ext p terms, the prove that (m + ) ( m p) m p m Solutio Let the A.P. be a, a + d, a + d,.... We are give a 1 + a a m a m+1 + a m a m+... (1) Addig a 1 + a a m o both sides of (1), we get [a 1 + a a m ] a 1 + a a m + a m a m+ S m S m+ m m Therefore, a( m1) d a( m1) d Puttig a + (m 1) d x i the above equatio, we get mx m (x + d) (m m ) x (m + ) d (m ) x (m + ) d... () Similarly, ifa 1 + a a m a m a m a m + p Addig a 1 + a a m o both sides we get, (a 1 + a a m ) a 1 + a a m a m + p or, S m S m + p m m p { a ( m 1) d } {a + (m + p 1)d} which gives i.e., (m p) x (m + p)pd... (3) Dividig () by (3), we get

9 SEQUENCE AND SERIES 155 ( m) x ( m) d ( m p) x ( m p) pd (m ) (m + p) p (m p) (m + ) Dividig both sides by mp, we get (m + p) (m + ) 1 1 m (m + ) 1 1 p m 1 1 m p (m + p) 1 1 m Example 10 If a 1, a,..., a are i A.P. with commo differece d (where d 0); the the sum of the series si d (cosec a 1 cosec a + cosec a cosec a cosec a 1 cosec a ) is equal to cot a 1 cot a Solutio We have si d (cosec a 1 cosec a + cosec a cosec a cosec a 1 cosec a ) si d... si a1si a si asi a3 si a1si a si( a a1) si ( a3a) si ( aa 1)... si a si a si a si a si a si a sia cos a1 cos asi a1) si a3cos acos a3si a) sia cos a1cos a si a1)... sia sia sia sia sia si a (cot a 1 cot a ) + (cot a cot a 3 ) (cot a 1 cot a ) cot a 1 cot a Example 11 (i) If a, b, c, d are four distict positive quatities i A.P., the show that bc > ad (ii) If a, b, c, d are four distict positive quatities i G.P., the show that a + d > b + c Solutio (i) Sice a, b, c, d are i A.P., the A.M. > G.M., for the first three terms.

10 156 EXEMPLAR PROBLEMS MATHEMATICS Therefore, b > ac Here a c b Squarig, we get b > ac... (1) Similarly, for the last three terms AM > GM b d c > bd Here c c > bd Multiplyig (1) ad (), we get b c > (ac) (bd) bc > ad (ii) Sice a, b, c, d are i G.P. agai A.M. > G.M. for the first three terms a c > b sice ac b a + c > b Similarly, for the last three terms b d > c sice bd c b + d > c Addig (3) ad (4), we get (a + c) + (b + d) > b + c a + d > b + c... ()... (3)... (4) Eample 1 If a, b, c are three cosecutive terms of a A.P. ad x, y, z are three cosecutive terms of a G.P. The prove that x b c. y c a. z a b 1 Solutio We have a, b, c as three cosecutive terms of A.P. The b a c b d (say) c a d a b d

11 SEQUENCE AND SERIES 157 Now x b c. y c a. z a b x d. y d. z d x d d d ( xz). z (sice y ( xz )) as x, y, z are G.P.) x d. x d. z d. z d x d + d. z d d x z 1 Example 13 Fid the atural umber a for which f ( a k) 16( 1), where the fuctio f satisfies f (x + y) f (x). f (y) for all atural umbers x, y ad further f (1). Solutio Give that f (x + y) f (x). f (y) ad f (1) Therefore, f () f (1 + 1) f (1). f (1) f (3) f (1 + ) f (1). f () 3 f (4) f (1 + 3) f (1). f (3) 4 ad so o. Cotiuig the process, we obtai f (k) k ad f (a) a k1 Hece f ( a k) k1 k1 f (). a f () k f (a) k 1 f ( k) a ( ) 1. 1 a a ( 1) 1... (1) But, we are give f ( ak) 16 ( 1) k1 a + 1 ( 1) 16 ( 1) a+1 4 a a 3

12 158 EXEMPLAR PROBLEMS MATHEMATICS Objective Type Questios Choose the correct aswer out of the four give optios i Examples 14 to 3 (M.C.Q.). Example 14 A sequece may be defied as a (A) relatio, whose rage N (atural umbers) (B) fuctio whose rage N (C) fuctio whose domai N (D) progressio havig real values Solutio (C) is the correct aswer. A sequece is a fuctio f : N X havig domai N Example 15 If x, y, z are positive itegers the value of expressio (x + y) (y + z) (z + x) is (A) 8xyz (B) > 8xyz (C) < 8xyz (D) 4xyz Solutio (B) is the correct aswer, sice x y y z z x A.M. > G.M., xy, yz ad Multiplyig the three iequalities, we get xy yz yz.. ( xy)( yz)( zx) or, (x + y) (y + z) (z + x) > 8 xyz zx Example 16 I a G.P. of positive terms, if ay term is equal to the sum of the ext two terms. The the commo ratio of the G.P. is (A) si 18 (B) cos18 (C) cos 18 (D) si 18 Solutio (D) is the correct aswer, sice t t +1 + t + ar 1 ar + ar +1 1 r + r 1 5 r, sice r > 0 Therefore, r 5 1 si 18 4

13 SEQUENCE AND SERIES 159 Example 17 I a A.P. the pth term is q ad the (p + q) th term is 0. The the qth term is (A) p (B p (C) p + q (D) p q Solutio (B) is the correct aswer Let a, d be the first term ad commo differece respectively. Therefore, T p a + (p 1) d q ad... (1) T p+ q a + (p + q 1) d 0... () Subtractig (1), from () we get qd q Substitutig i (1) we get a q (p 1) ( 1) q + p 1 Now T q a + (q 1) d q + p 1 + (q 1) ( 1) q + p 1 q + 1 p Example 18 Let S be the sum, P be the product ad R be the sum of the reciprocals of 3 terms of a G.P. The P R 3 : S 3 is equal to (A) 1 : 1 (B) (commo ratio) : 1 (C) (first term) : (commo ratio) (D) oe of these Solutio (A) is the correct aswer a Let us take a G.P. with three terms, aar,. The r S a a( r r 1) a ar r r P a 3, R r 1 1 1r r1 a a ar a r 6 1 r r1 3 a 3 P R a r S 3 r r 1 a r Therefore, the ratio is 1 : 1 3 Example 19 The 10th commo term betwee the series ad is (A) 191 (B) 193 (C) 11 (D) Noe of these Solutio (A) is the correct aswer.

14 160 EXEMPLAR PROBLEMS MATHEMATICS The first commo term is 11. Now the ext commo term is obtaied by addig L.C.M. of the commo differece 4 ad 5, i.e., 0. Therefore, 10 th commo term T 10 of the AP whose a 11 ad d 0 T 10 a + 9 d (0) 191 Example 0 I a G.P. of eve umber of terms, the sum of all terms is 5 times the sum of the odd terms. The commo ratio of the G.P. is 4 1 (A) (B) (C) 4 (D) oe the these 5 5 Solutio (C) is the correct aswer Let us cosider a G.P. a, ar, ar,... with terms. We have ar ( 1) 5 a ( r ) 1 r 1 r 1 (Sice commo ratio of odd terms will be r ad umber of terms will be ) ar ( 1) ar ( 1) 5 r1 ( r 1) a (r + 1) 5a, i.e., r 4 Example 1 The miimum value of the expressio 3 x x, x R, is 1 (A) 0 (B) 3 Solutio (D) is the correct aswer. We kow A.M. G.M. for positive umbers. (C) 3 (D) 3 Therefore, x x x 3 3 1x x 1 x 3 3 x 3 3 x 3 3 x x 3

15 SEQUENCE AND SERIES EXERCISE Short Aswer Type 1. The first term of a A.P.is a, ad the sum of the first p terms is zero, show that a( pq) q the sum of its ext q terms is. [Hit: Required sum S p + q S p ] p 1. A ma saved Rs i 0 years. I each succeedig year after the first year he saved Rs 00 more tha what he saved i the previous year. How much did he save i the first year? 3. A ma accepts a positio with a iitial salary of Rs 500 per moth. It is uderstood that he will receive a automatic icrease of Rs 30 i the very ext moth ad each moth thereafter. (a) Fid his salary for the teth moth (b) What is his total earigs durig the first year? 4. If the pth ad qth terms of a G.P. are q ad p respectively, show that its (p + q) th term is q p p q 1 pq. 5. A carpeter was hired to build 19 widow frames. The first day he made five frames ad each day, thereafter he made two more frames tha he made the day before. How may days did it take him to fiish the job? 6. We kow the sum of the iterior agles of a triagle is 180. Show that the sums of the iterior agles of polygos with 3, 4, 5, 6,... sides form a arithmetic progressio. Fid the sum of the iterior agles for a 1 sided polygo. 7. A side of a equilateral triagle is 0cm log. A secod equilateral triagle is iscribed i it by joiig the mid poits of the sides of the first triagle. The process is cotiued as show i the accompayig diagram. Fid the perimeter of the sixth iscribed equilateral triagle. 8. I a potato race 0 potatoes are placed i a lie at itervals of 4 metres with the first potato 4 metres from the startig poit. A cotestat is required to brig the potatoes back to the startig place oe at a time. How far would he ru i brigig back all the potatoes? 9. I a cricket touramet 16 school teams participated. A sum of Rs 8000 is to be awarded amog themselves as prize moey. If the last placed team is awarded

16 16 EXEMPLAR PROBLEMS MATHEMATICS Rs 75 i prize moey ad the award icreases by the same amout for successive fiishig places, how much amout will the first place team receive? 10. If a 1, a,..., a are i A.P., where a i > 0 for all i, show that a a a a a a a a Fid the sum of the series (3 3 3 ) + ( ) + ( ) +... to (i) terms (ii) 10 terms 1. Fid the r th term of a A.P. sum of whose first terms is + 3. [Hit: a S S 1 ] Log Aswer Type 13. If A is the arithmetic mea ad G 1, G be two geometric meas betwee ay two umbers, the prove that 1 G G A G G If θ 1, θ, θ 3,..., θ are i A.P., whose commo differece is d, show that ta secθ 1 secθ + secθ secθ secθ 1 secθ ta 1. si d 15. If the sum of p terms of a A.P. is q ad the sum of q terms is p, show that the sum of p + q terms is (p + q). Also, fid the sum of first p q terms (p > q). 16. If p th, q th, ad r th terms of a A.P. ad G.P. are both a, b ad c respectively, show that Objective Type Questios a b c. b c a. c a b 1 Choose the correct aswer out of the four give optios i each of the Exercises 17 to 6 (M.C.Q.). 17. If the sum of terms of a A.P. is give by S 3 +, the the commo differece of the A.P. is (A) 3 (B) (C) 6 (D) 4

17 SEQUENCE AND SERIES The third term of G.P. is 4. The product of its first 5 terms is (A) 4 3 (B) 4 4 (C) 4 5 (D) Noe of these 19. If 9 times the 9 th term of a A.P. is equal to 13 times the 13 th term, the the d term of the A.P. is (A) 0 (B) (C) 0 (D) If x, y, 3z are i A.P., where the distict umbers x, y, z are i G.P. the the commo ratio of the G.P. is 1 1 (A) 3 (B) (C) (D) 3 1. If i a A.P., S q ad S m qm, where S r deotes the sum of r terms of the A.P., the S q equals (A) 3 q (B) mq (C) q 3 (D) (m + ) q. Let S deote the sum of the first terms of a A.P. If S 3S the S 3 : S is equal to (A) 4 (B) 6 (C) 8 (D) The miimum value of 4 x x, x R, is (A) (B) 4 (C) 1 (D) 0 4. Let S deote the sum of the cubes of the first atural umbers ad s deote the sum of the first atural umbers. The (A) ( 1)( ) 6 (B) Sr s equals r1 r ( 1) 3 (C) (D) Noe of these 5. If t deotes the th term of the series the t 50 is (A) 49 1 (B) 49 (C) (D) The legths of three uequal edges of a rectagular solid block are i G.P. The volume of the block is 16 cm 3 ad the total surface area is 5cm. The legth of the logest edge is (A) 1 cm (B) 6 cm (C) 18 cm (D) 3 cm

18 164 EXEMPLAR PROBLEMS MATHEMATICS Fill i the blaks i the Exercises 7 to For a, b, c to be i G.P. the value of ab bc is equal to The sum of terms equidistat from the begiig ad ed i a A.P. is equal to The third term of a G.P. is 4, the product of the first five terms is.... State whether statemet i Exercises 30 to 34 are True or False. 30. Two sequeces caot be i both A.P. ad G.P. together. 31. Every progressio is a sequece but the coverse, i.e., every sequece is also a progressio eed ot ecessarily be true. 3. Ay term of a A.P. (except first) is equal to half the sum of terms which are equidistat from it. 33. The sum or differece of two G.P.s, is agai a G.P. 34. If the sum of terms of a sequece is quadratic expressio the it always represets a A.P. Match the questios give uder Colum I with their appropriate aswers give uder the Colum II. 35. Colum I Colum II (a) 4, 1, 1 4, 1 16 (i) A.P. (b), 3, 5, 7 (ii) sequece (c) 13, 8, 3,, 7 (iii) G.P. 36. Colum I Colum II (a) (i) ( 1) (b) (ii) ( + 1) (c) (iii) (d) (iv) ( 1)(1) 6 ( 1)