Lesson-2 PROGRESSIONS AND SERIES

Transcription

1 Lesso- PROGRESSIONS AND SERIES Arithmetic Progressio: A sequece of terms is sid to be i rithmetic progressio (A.P) whe the differece betwee y term d its preceedig term is fixed costt. This costt is clled the commo differece (c.d) of the A.P. If is first term d d is the commo differece of the A.P., the its th term t is give by t + ( )d. The sum S of the first terms of such A.P. is give by S [ + ( )d] [ + l] where l is the th term of the A.P. Importt results o A.P.. If, b, c re i A.P.d k is o zero costt the, () + k, b + k, c + k re i A.P. k, bk, ck re i A.P.. If,,,... d b, b, b,... re two A.P s, the + b, + b, + b,... is lso A.P.. Three terms i A.P. my be tke s d,, + d. 4. Four terms i A.P. my be tke s d, d, + d, + d. 5. Arithmetic Me (A.M.) If, A, b re i A.P. the, A is sid to be the rithmetic me of d b. Thus, A b If,,,..., re umbers, the the rithmetic me A of these umbers is give by A ( ) The umbers A, A, A,..., A re sid to be the rithmetic mes iserted betweee d b, if, A, A, A,..., A, b re i A.P. Let d be the commo differece of this A.P.; the, b + ( + )d d d b A k + k ( b ) : k,...,

2 Geometric Progressio (G.P.) A sequece is sid to be i G.P. whe its first term is o-zero d ech term is r times the preceedig term,where r is o-zero costt. The fixed umber r is kow s the commo rtio of the G.P. The th term t is give by t r where is the first term. Sum of first terms of G.P. is give by S ( r ), if r r S + r + r + r +... r : < r <, r 0 Some results for G.P.. If, b, c re i G.P., the for k 0 () k, bk, ck re i G.P. b c,, re i G.P. k k k. Three terms i G.P my be tke s r,, r. Four terms i G.P., my be tke s,, r, r r r 4. The geometric me G of two positive umbers d b is give by b. If,,,..., re positive umbers, the their geometric me is give by (... ) / 5. If,,,... ( i > 0 i) re i G.P. the log, log, log,... re i A.P. d vice vers. Geometric mes betewee two umbers G, G, G,..., G re clled geometric mes iserted betwee two umbers d b, if, G, G, G,..., G, b is i G.P. Let r be the commo rtio of this G.P.; the b r + ; b r Hece G k r k b, ( k,,,..., ). k

3 Hrmoic Progressio A sequece,,,... is sid to be i hrmoic progressio (H.P.), if i rithmetic progressio.,,,... re The th term of the H.P. is give by ( )d where d d If d b re two o-zero umbers, the the hrmoic me of d b is umber H such tht the sequece, H, b is i H.P. We hve b or H H b b If,,,..., re o-zero umbers, the the hrmoic me H of these umbers is give by H... Hrmoic mes betwee two umbers H, H, H,..., H re hrmoic mes iserted betwee two umbers d b, if, H, H, H,..., H, b re i H.P. Thus,,, H H,, H..., H, b re i A.P. with commo differece d. b + [( + ) ]d ; d b b ( ) b ; b kd k : k,..., ( ) b H k H k b( ) b( ) k( b), (k,,,..., ) Reltio mog A, G, H If A, G d H deote the A.M., G.M. d H.M. respectively betwee two positive umbers d b, the A b, G b, H b b AH. b G b AM, G.M., H.M. re i G.P. b b

4 b b Further, A G b 0 Similrly G H. A G H. Arithmetic Geometric Series b A series i which ech term is the product of correspodig terms i rithmetic d geometric progressio, is clled Arithmetic Geometric Series. Thus, from the A.P., + d, + d,... d the G.P., r, r,... We get the rithmetico geometric series + ( + d)r + ( + d)r +... th term of Arithmetic Geometric series T [ + ( )d ]r - Sum of first terms of Arithmetic Geometric series S + ( + d)r + ( + d)r + ( + d)r +... [ + ( )d]r rs r + ( + d)r + ( + d)r +... [ + ( )d]r + [ + ( )d]r By subtrctio d solvig, we get b S ( r dr r ( r) ) [ ( ) d] r r Sum of ifiite Arithmetic Geometric series Whe r <, Stdrd Results S dr r ( r ). Sum of the first turl umbers r r ( ). Sum of squres of the first turl umbers r. Sum of cubes of the first turl umbers r r r ( )( ) ( )

5 SOLVED EXAMPLES Ex.: How my terms re there i the A.P. 0, 5, 0, 5,... 00? Sol.: Ex.: Sol.: Ex.: Sol.: Let the umber of terms be t 00, 0, d 5 t + ( )d ( )5 80 ( )5 or 6 ( ) 7 If the pth term of A.P. is q d the qth term is p the fid the rth term. Let the iitil term d commo differece of the A.P. be deoted by d d respectively. + (p )d q...(i) + (q )d p...(ii) Solvig (i) & (ii), we get q p (p q)d d Substitutig d i (i) we get q + p t r + (r )d (q + p ) r + p + q r Fid the sum of the series to 0 terms The terms of give series re i A.P. d 4, 99 d 0. 0 S 0 [ 99 + (0 ) ( 4)] 0 [ ( 4)] 0 (98 76) 0. Ex.4: If S, S, S be the sum of first terms of rithmetic progressios, the first term of ech beig d the commo differeces beig, d respectively, the show tht : S + S S Sol.: Here, d, d, d S [ + ( )d ] [ + ( )] [ + ] S [ + ( )d ] [ + ( )]

6 S [ + ( )d ] [ + ( )] [ ] S + S [ + + ] S Ex.5: Fid four terms i A.P. such tht their sum is 50 d gretest of them is 4 times the lest Sol.: Let the four umbers i A.P. be d, d, + d, + d with d > 0. ( d) + ( d) + ( + d) + ( + d) ; 5 + d 4 ( d) d 5 5 Hece the four umbers re 5, 0, 5, 0. Ex.6: If the fifth term of G.P. is 8 d secod term is 4, fid the G.P. Sol.: t r t 5 8 d t 4 8 r 4...(i) d 4 r...(ii) Dividig (i) by (ii), we get 8 4 r or r Puttig the vlue of r i (ii), we get, 6 Hece the required G.P. is 6, 4, 6, 54,... r Ex.7: Fid the sum to terms of the series : Sol.: Let S to terms 8 [ ] 9 8 [ ] 9 8 [(0 ) + (00 ) + (000 ) +... to terms] 8 8 [ to terms ] ( 0 ) [ ] 8

7 Ex.8: If S, S, S,..., S p re the sum of p ifiite geometric progressios whose first terms re,,,..., p d whose commo rtios re S + S S p p( p ),,, 4... respectively, prove tht p Sol.: We kow S S r S S p p p p + S + S S p (p + ) p [. + (p ).] p [p +] Ex.9: Fid S Sol.: S (i) S (ii) Substrctig (ii) from (i) S S 00 S Ex.0: If the 7th term of H.P. is 8 d the 8th term is 7, the fid its 5th term. Sol.: t 7 8 d t 8 7 ; Let the correspodig A.P. be, + d,... 6d 8, Solvig, d 56 7d 7 t 5 4d 56 5

8 Ex.: Sol.: The sum of the terms of two A.P. s re i the rtio : Fid the rtio of their th terms. Let, be the first terms of two A.P s d d, d be their respective commo differeces. [ [ ( ) d] 5 4 ( ) d ] 9 6 d d The rtio of the th terms d 9. d d d...(i) (which is obtied from (i) with 5) Ex.: Suppose x, y, z re positive rel umbers which re differet from. If x 8 y z 8, show tht, log y (x), log z (y) d 7log x (z) re i A.P. Sol.: x 8 y z 8, 8 log x log y 8 log z log x 7 log y x log y 6 log y x 7...(i) log z y log y log z 4 log z y 4...(ii) log x z log z log x log x z 9...(iii) But the umbers, 7, 4, 9 re i A.P with commo differece, log y x, log z y d 7 log xz re i A.P. Ex.: If,,,..., re i A.P., where i > 0 for ll i, show tht... Sol.:,,,..., re i A.P.... d (sy)

9 )( ) )( )... ( L.H.S. ( )( ) d ( ( ) d,... d d ( ) d + ( )d d ( ) L.H.S. ( ) R.H.S. d Ex.4: Determie the reltio betwee x, y d z if, log y x, log z y, 5log x z re i A.P. Sol.: Let d be the commo differece of the give A.P.; the, log y x + d x y +d...(i) log z y + d y z +d...(ii) d 5 5log x z + d z x Elimitig y & z from (i), (ii) d (iii), we get...(iii) (d )( d )( d ) x x 5 ( d)( d)( d ) 5 or ( + d)( + d)( + d) or (d + )(6d d + 8) 0 d The other fctors re x y, y z, z x / or x y z

10 Ex.5: If x y z b c d if, b, c re ll positive d i G.P., the prove tht x, y, z re i A.P. Sol.: x b y c z log x logb y logc z k log kx, log b ky, log c kz...(i), b, c re i G.P. b c log b log + log c ky kx + kz by (i) y x + z x, y, z re i A.P. Ex.6: If pth, qth d rth terms of both A.P. d G.P. re, b, c respectively, prove tht b c. bc. c b (both progressios hve the sme first term). Sol.: t p + (p )d (r ) p t q b + (q )d (r ) q t r c + (r )d (r ) r b c. bc. c b ( r p ) b c ( r q ) c ( r r ) b b c+c + b. r (p )(b c)+(q )(c )+(r )( b) Ex.7: There re four umbers of which the first three re i G.P. d the lst three re i A.P. with commo differece 6. If the first umber d the lst umber re equl, fid the umbers. Sol.: Let the four umbers be, d, d, where d 6,, 6 re i G.P. ( 6) ( ) Numbers re 8, 4,, 8 Ex.8: The series of turl umbers is divided ito groups : () ; (,, 4) ; (5, 6, 7, 8, 9) d so o. Show tht the sum of the umbers i the th group is ( ) +. Sol.: The lst term of ech group is the squre of the correspodig umber of the group. Hece the first term of the th group is ( ) + + Number of terms i the first group Number of terms i the secod group Number of terms i the third group 5

11 No. of terms i the th group Commo differece of the umbers i the th group Required Sum [( + ) + ( )] [ + ] ( )[ + ] + + ( ) Ex.9: Sum the series 5... to Sol.: S... to S + S...(i) 5 where S... to S to Subtrctig, 4 S... to From (i), S 8 S 8 8

12 Ex.0: If N d >, prove tht () ( ) + Sol.: () Sice A.M G.M, we hve 5... ( ) [ ( )] / ( ) [ ( )] / () ( ).... (..... ) / ( ) + /

13 OBJECTIVE QUESTIONS Choose the correct optio(s) i the followig. The sum of p terms of A.P. is q d the sum of q terms is p. The sum of p + q terms is () p + q p q 0 (p + q) bx b cx c dx. If ( x 0), the, b, c d d re i bx b cx c dx () A.P. G.P. H.P. oe of these. If there re 4 hrmoic mes betwee d 4 the the third hrmoic me is () If,,, 4, 5 re i H.P. the is equl to () oe of these 5. If, b, c re i G.P. the log x, log b x, log c x re i () AP GP HP oe of these 6. If, b, c re i A.P. d G.P., the () b c b c b c b c 7. If the p th term of H.P. be qr d q th term be rp the r th term is () pq 0 I (p + q) 8. If, x, y, z, re i G.P. the xyz () 4 8 oe of these 9. If, b, c, d, e, f re the A.M s betwee d, the + b + c + d + e + f () oe of these 0. If 0 < <, the t + cot is () < c hve y vlue oe of these. Let T r be the rth term of A.P., for r,,,.... If, for some positive iteger m,, we hve T m d T, the T m m equls

14 () m oe of these m. For y odd iteger, ( ) ( ) is equl to () ( )( ) 4 ( )( ) 4 [ ( )] 4 oe of these. If cos(x y), cos x d cos(x + y) re i H.P., the cos x sec y () oe of these 4. The hrmoic me of the roots of the equtio ( 5 ) x (4 5 ) x is equl to () If x b y c z d t d, b, c d d re i G.P., the x, y, z re i () A.P. G.P. H.P. oe of these 6. If x >, y >, z > re i G.P., the,, l x l y l z re i () A.P. G..P. H.P. oe of these 7. If the first d ( )th terms of A.P., G.P d H.P re equl d their th terms re, b d c respectively, the () b c + c b > b > c oe of these 8. Let,,,..., 0 be i A.P. d h, h, h,..., h 0 be i H.P. If h d 0 h 0, the 4 h 7 is () 5 oe of these The sum of first terms of the series... is equl to () + oe of these 0. If, b, c, d re positive rel umbers such tht + b + c + d, the M ( + b )( c + d ) stisfies the reltio () 0 < M M M oe of these. The product of positive umbers is uity. The their sum is () positive iteger divisible by equl to + / oe of these

15 . If log (5. x + ), log 4 ( x + ) d re i A.P., the x is equl to () log 5 log 0.4 log 5 oe of these. The sum of terms of the series + ( + ) + ( + + 5) +... is () ( ) ( )( ) 6 oe of these 4. If,,,... is A.P. such , the is equl to () oe of these 5. I G.P. of positive terms, y term is equl to the sum of the ext two terms. The the commo rtio of the G.P. is () cos 8º si 8º cos 8º oe of these 6. If, b, c re i G.P. d + x, b + x, c + x re i H.P., the the vlue of x is, (, b, c re distict umbers). () c b oe of these 7. The sum of terms of the series... is give by () ( + ) oe of these 8. The qurdtic equtio i x such tht the rthmtic me is A d their geometric me is G is () x Ax + G 0 x + Ax + G 0 x Ax G 0 x + Ax G The sum of terms of the series +... is 4 ( ) () ( ) ( ) ( ) ( + ) oe of these 0. Lim t is r r () 6 4 MORE THAN ONE CORRECT ANSWERS. If b, b, b (b > 0) re three successive terms of G.P. with commo rtio r, the vlue of r for which the iequlity b > 4b b holds is give by () r > r < r.5 r 5.

16 . Let the hrmoic me d the geometric me of two positive umbers be i the rtio 4 : 5, the two umbers re i the rtio () 4: :4 5:4 oe of these.. If log x, x/ x d log b re i G.P. the x is equl to () log (log b ) log (log e ) log (log e b) log (log b) oe of these 4. If A, A : G, G ; d H, H re two AM s, GM s d HM s respectively, betwee two qutities d b, the b is equl to () A H A H G G oe of these. 5. If x + 9y + 5z xyz 5 5 x y z, the () x, y d z re i HP,, x y z re i AP x, y, z re i GP,, x y z re i GP ( c b bc)( b bc c) 6. If, b,c re i HP, the vlue of ( bc) is ( c)( c) () 4 c bc b ( c)( c) bc 4 c 7. If r( r )(r ) 4 + b + c + d + e, the r () b d c e 0, b /, c re i AP ( b d) is iteger 8. If, b, c re i Gp d x d y respectively, be rithmtic mes betwee, b d b,c the c () x y c c x y x y b x y c 9. If >, the vlues of the positive iteger m for which m + divides is/re () Let E..., the () E < E > / E > E <

17 MISCELLANEOUS ASSIGNMENT Comprehesio- Recll the method of defferece i the summtio of miscelleous series. For exmple if th term for series be, T the we c express it s differece of two terms. ( ) ( ) ( ) s T ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) V V the S (V V 0 ) + (V V ) (V V ) (V V 0 ). Usig bove iformtio solve the followig questios. The sum of the series upto 00 terms is () oe of these. If,,... re i A. P. with commo differece d, the the sum of the series si d [cosec cosec + cosec cosec cosec cosec ] is () sec sec cosec cosec cot cot t t. The sum of terms of the series is S, the the vlue of...4 S is equl to () 0 does t exist 4. The sum to terms of the series is () 5. The sum of the series si x [si x + si 5x + si 7x si ( + )x] is () cos x cos ( + )x cos x cos (x + ) x cos x + cos ( + ) x oe of these

18 Comprehesio- The ifiite series + x + x + x +... ( x < ) hs the sum x 4x +... c be dded with the help of previous oe. Let us study this. If S + x + x + 4x +... the xs x + x + x +... S( x) + x + x + x +... S ( x) x ( x < ) We ote tht coefficiet of x i the expsio of ( x) is +.. The ifiite series + x + x + The ifiite series my be used to fid the umber of wys of distributig ideticl objects ito distict boxes. For exmple, the umber of wys i which 5 ideticl blls c be distributed i differet boxes must be equl to coefficiet of x 5 i the ifiite product (x 0 + x + x +...) (x 0 + x + x +...) (x 0 + x + x +...). For y wy i which x 5 is obtied is oe of the wys i which 5 ideticl blls c be distributed i distict boxes. While fidig the umber of wys the coditio x < my be isigifict. 6. The coefficiet of y i the expressio y y must be () oe of these 7. The sum of series + x + 6x + 0x + 5x ( x < ) must be () ( x) x x oe of these 8. The coefficiet of x i the expsio of ( x) must be () ( ) ( ) ( )( ) oe of these 9. The umber of wys i which we c get totl of 0 while throwig dice must be () MATCH THE FOLLOWING: 0. Mtch the followig if, b, c, re i H.P., the A. b c,, b c c b b c (p) H.P. B.,, b b b c (q) G.P. C. b, b, c b (r) A.P. D. b c,, b c c b

19 . A. If, b, c re i G.P., the log 0, log b 0, log c 0 re i (p) A.P. B. If x x x be b ce c de be b ce c de x x x, the, b, c re i (q) H.P. C. If, b, c, d re i AP;, x, b re i G.P., b, y, c re i G.P., the x, b, y re (r) G.P. D. If x, y, z re i G.P., x b y c z, the log, log b, log c re i (s) oe of these INTEGER TYPE QUESTIONS. Fid the umber of solutio i [0, ] which stisfyig the equtio 8 x x x 4 cos cos cos The sum of ifiite terms of the series If S deotes the sum of first -terms of A.P. if S S, the the rtio of S /S is equl to 5. If H d the vlue of S... is of the form H the vlue of is 6. If, b, c d d re distict itegers i A.P. such tht d + b + c, the + b + c + d 7. x + y + z 5 whe, x, y, z, b re i A.P. 5 whe, x, y, z, b re i H.P. the b x y z 8. Let 6, 4,,... be geometric sequece. Defie P s the product of first terms. The the P vlue of 4 is If S /( 5) + /( 5 7) + /(5 7 9) +..., the the vlue of [6 S]. (where [.] represets the gretest iteger fuctio) is Betwee d, m rithmetic mes re iserted so tht the rtio of the seveth d (m ) th mes is 5 : 9. The the vlue of m/ is.... If the cotiued product of three umbers i G.P. is 6 d the sum of their products i pirs is 56, the the commo rtio c be...

20 PREVIOUS YEAR QUESTIONS IIT-JEE/JEE-ADVANCE QUESTIONS. If, b, c re i G.P., the the equtios x + bx + c 0 d dx + ex + f 0 hve commo root if d e f,, re i b c () A.P. G.P. H.P. oe of these 7 5. Sum of the first terms of the series... is equl to () +.. If the first d ( ) th terms of A.P., G.P. d H.P. re equl d their th terms re, b d c respectively, the () b c b 0 + c b c b 0 4. Let T r be the r th term of A.P., for r,,,... if for some positive itegers m, we hve T m d T m, the Tm equls () m 0 m 5. If x >, y >, z > re i G.P., the,, re i l x l y l z () A.P. H.P. G.P. oe of these 6. Let,,..., 0 be i A.P. d h, h,..., h 0 be i H.P. If h d 0 h 0, the 4 h 7 is () Cosider ifiite geometric series with first term d commo rtio r. If its sum is 4 d the secod term is 4, the () 4 7, r 7, r 8, r, r 4 8. If the sum of the first terms of the A.P., 5, 8,..., is equl to the sum of the first terms of the A.P. 57, 59, 6,..., the equls () 0 9. Let the positive umbers, b, c, d be i A.P. The bc, bd, cd, bcd re () ot i A.P./G.P./H.P. i A.P. i G.P. i H.P.

21 0. If,,..., re positive rel umbers whose product is fixed umber c, the the miimum vlue of is () / ( + )c / c / ( + ) /. Suppose, b, c re i A.P. d, b, c re i G.P. If < b < c d + b + c, the the vlue of is () tht b > ( ) d b 4 the the miimum vlue of 0 such Prgrph for Questio Nos. to 5 Let V r deote the sum of the first r terms of rithmetic progressio (A.P.) whose first term is r d the commo differece is (r ). Let T r V r+ V r d Q r T r+ T r for r,,.... The sum V + V V is () ( + )( + ) ( + )( + + ) ( + ) ( + ) 4. T r is lwys prime umber composite umber 5. Which oe of the followig is correct sttemet? () Q, Q, Q,... re i A.P. with commo differece 5 Q, Q, Q,... re i A.P. with commo differece 6 Q, Q, Q,... re i A.P. with commo differece Q Q Q... Prgrph for Questio Nos. 6 to 8 Let A, G, H deote the rithmetic, geometric d hrmoic mes, respectively, of two distict positive umbers. For, let A d H hve rithmetic, geometric d hrmoic mes s A, 6. Which oe of the followig sttemets is correct? () G > G > G > G < G < G < G G G G < G < G 5 < d G > G 4 > G 6 > 7. Which oe of the followig sttemets is correct? () A > A > A > A < A < A <

22 A > A > A 5 > d A < A 4 < A 6 < A < A < A 5 < d A > A 4 > A 6 > 8. Which oe of the followig sttemets is correct? () H > H > H > H < H < H < H > H > H 5 > d H < H 4 < H 6 < H < H < H 5 < d H > H 4 > H 6 > 9. A stright lie through the vertex P of trigle PQR itersects the side QR t the poit S d the circumcircle of the trigle PQR t the poit T. If S is ot the cetre of the circumcircle, the () PS ST QS SR PS ST QS SR PS ST 4 QR PS ST 4 QR 0. Suppose four distict positive umbers,,,, 4 re i G.P. Let b, b b +, b b + d b 4 b + 4. STATEMENT-: The umbers b, b, b, b 4 re either i A.P. or i G.P. STATEMENT-: The umbers b, b, b, b 4 re i H.P. () Sttemet- is True, Sttemet- is True; Sttemet- is correct expltio for Sttemet- Sttemet- is d is True; Sttemet- is NOT correct expltio for Sttemet- Sttemet- is True, Sttemet- is Flse Sttemet- is Flse, Sttemet- is True. If the sum of first terms of A.P. is c, the the sum of squres of these terms is () (4 ) c 6 (4 ) c (4 ) c (4 ) c 6. Let S k, k,,...00, deote the sum of the ifiite geometric series whose first term is commo rtio is 00 k. The the vlue of 00 ( k k ) Sk is 00! k. Let,,,..., be rel umbers stisfyig 5, 7 > 0 d k k k for k, 4,..., k k! d the If... 90, the the vlue of... is equl to 4. The miimum vlue of the sum of rel umbers 5, 4,,, 8 d 0 with > 0 is

23 5. Let,,,, 00 be rithmetic progressio with d S, p 00. For y p p i i iteger with 0, let m 5. If S S m does ot deped o, the is 6. Let,,... be i hrmoic progressio with 5 d 0 5. The lest positive iteger for which < 0 is () 4 5 Prgrph for Questio 7 to 8 Let deote the umber of ll -digit positive itegers formed by the digits 0, or both such tht o cosecutive digits i them re 0, Let b the umber of such -digit itegers edig with digit d c the umber of such -digit itegers edig with digit The vlue of b 6 is () Which of the followig is correct? () k 9. Let k S 4 k k. The S c tke vlue(s) () A pck cotis crds umbered from to. Two cosecutive umbered crds re removed from the pck d the sum of the umbers o the remiig crds is 4. If the smller of the umbers o the removed crds is k, the k 0. Let, b, c be positive itegers such tht b is iteger. If, b, c re i geometric progressio d the rithmetic mes of, b, c is b +, the the vlue of is DCE QUESTIONS. If p, q, r re i A P the p th, q th d r th terms of y GP re i () A. P. G.P. reciprocls of these terms re i A.P. oe of these. log, log 6, log re i () A.P. G.P. H.P. oe of these

24 . The sum of series, 5, 8,,... is 6000, the is () The rtiol umber which is equl to the umber. 57 with recurrig deciml is () oe of these 5. If, b, c re i A.P., the x +, bx +, cx +, x 0 re i () A.P. G.P. oly whe x > 0 G.P. if x < 0 G.P. for ll x 0 6. bc c b,, re i H.P. The c b bc b bc c (),, b c re i H.P., b, c re i H.P. bc, c, b re i A.P. (c + b), (bc + b), (bc + c) re i A.P. 7. If, b, c re i G.P., the the equtios x + bx + c 0 d x + ex + f 0 hve commo root if d e f,, re i b c () A. P. G. P. H. P. oe of these 8. If G d G re two GM s d A the AM s iserted betwee two umbers, the the vlue of G G is G G A () A A oe of these 9. If x b y c z d x, y, z re i H.P., the, b, c re i () A.P. H.P. G.P. oe of these is equl to () 6 9. If y + x + x + x +..., the x () y y y y y y oe of these. If, b, c re i G. P. d log c, log b c, log b re i A.P., the commo differece of A.P. is ()

25 . If A, A be two AM s d G, G be two GM s betwee d b, the to A A G G is equl () b b b b b b b b 4. The vlue of /4. 4 /8. 8 /6. 6 /... is () 5 5. Sum of the first terms of the series is equl to () ( ) The sum to ifiity of the series is () The sum of the 0 terms of the is () ( 6 ) 8. The sum of the first terms of the series 4 4 ( ) 4 ( ) is equl to 6 () + 9. If , the the vlue of x is equl to () 4 5 oe of these 0. + ( + ) + ( + + ) +... up to d term is () oe of these. If p th term of A.P. is q d q th term is p, the r th terms () q p + r p q + r p + q + r p + q r. If, b, c re i A.P. d, b, c re i H.P., the () b c + b c b + c c + b. If umbers p, q, r re i A.P., the m 7p, m 7q, m 7r (m > 0) re i () A.P. G.P. H.P. Noe of these

26 4. If, b, c, d d p re distict rel umbers such tht ( + b + c )p (b + bc + cd) p + (b + c + d ) 0 the, b, c, d re i () A.P. G.P. H.P. b cd 5. If log 0, log 0 ( x + ) d log 0 ( x + ) re three cosecutive terms of A.P., the () x 0 x x log 0 x log The sum of the series upto -terms is () 7. If S, S, S the the vlue of S lim S S 8 is equl to () / /64 9/ 9/64 8. If A.M. of two umbers is twice of their G.M. the the rtio of gretest umber to smllest umber is () AIEEE/JEE-MAINS QUESTIONS. If,,..., re i H.P., the the expressio : is equl to () ( ) ( ) ( )( ). If x 0, y b 0, z c where, b, c re i A.P. d <, b <, c < the x, y, z 0 re i () Arithmetic Geometric Progressio HP GP AP. Let T r be the rth term of A.P. whose first term is d commo differece is d. If for some positive itegers m,, m, T m d T, the d equls m () 0 m m

27 4. The sum of the first terms of the series is whe is eve. Whe is odd the sum is ( ) () ( ) ( ) ( ) 4 ( ) 5. If x, x, x d y, y, y re both i G.P. with the sme commo rtio, the the poits (x, y ), (x, y ) d (x, y ) () re vertices of trigle lie o stright lie lie o ellipse lie o circle. 6. The vlue of /4. 4 /8, 8 / is () / 4 7. Fifth term of G.P. is, the the product of its 9 terms is () oe of these 8. Sum of ifiite umber of terms of G. P. is 0 d sum of their squre is 00. The commo rtio of G. P. is () 5 /5 8/5 / () If, b, c re distict +ve rel umbers d + b + c the b + bc + c is () less th equl to greter th y rel o.. Let,,,... be terms of A.P. If () p p... q q 7, p q the 6 equls 7. The sum of the series... upto ifiity is!! 4! () e / e +/ e e. I geometric progressio cosistig of positive terms, ech term equls the sum of the ext two terms. The the commo rtio of this progressio equls () 5 ( 5 ) ( 5) 5 4. If p d q re positive rel umbers such tht p + q, the the mximum vlue of (p + q) is ()

28 The sum to ifiity of the series is 4 () A perso is to cout 4500 currecy otes. Let deote the umber of otes he couts i the th miute. If d 0,,... re i AP with commo differece, the the time tke by him to cout ll otes is () 5 miutes 4 miutes 4 miutes 5 miutes 7. A m sves Rs. 00 i ech of the first three moths of his service. I ech of the subsequet moths his svig icreses by Rs. 40 more th the svig of immeditely previous moth. His totl svig from the strt of service will be Rs. 040 fter : () 8 moths 9 moths 0 moths moths 8. The sum of first 0 terms of the sequece 0.7, 0.77, 0.777,..., is 7 0 () Three positive umbers form icresig G.P. If the middle term i this G.P. is doubled, the ew umbers re i A.P. The the commo rtio of the G.P. is () 0. If (0) 9 + () (0) 8 + () (0) () 9 k (0) 9, the k is equl to 8 9 ()

29 . I G.P. sum of terms is 64. First term is d commo rtio is. Fid.. Sum upto terms the series () The sum of ifiite geometric progressio is d the sum of the geometric progressio mde from the cubes of this ifiite series is 4. The fid the series 4. The sum of three umbers i A.P. is d the sum of their cubes is 88. Fid the umbers. 5. If, b, c re i A.P., prove tht () b + c, c +, + b re lso i A.P., bc, c BASIC LEVEL ASSIGNMENT re lso i A.P. b (b + c), b (c + ), c ( + b) re lso i A.P. b, c b c, c b re lso i A.P. 6. Fid the sum of the itegers betwee d 00 which re () multiple of multiple of 7 multiple of d 7 7. If S, S, S,... S p be the sums of first terms of p give A.P. s whose first terms re, respectively,,,,... d c.d. s re,,... prove tht S + S + S S p p ( + ) (p + ) 4 8. The sum of first terms of two A.P. s re i the rtio ( ) : (5 + ). Fid the rtio of their 4th terms. 9. If the pth term of A.P. is x d qth term is y, show tht the sum of first (p + q) terms is p q x y x y p q 0. Fid the sum to terms of the series If, b, c re i H.P. prove tht b c b c,, re lso i H.P. c b. Let, b, c, d, e be five rel umbers such tht, b, c re i A.P.; b, c, d re i G.P. c, d, e re i H.P. If d e 8, fid ll possible vlues of b, c d d.. Fid the () sum of first terms of the series : sum of first terms of the series :

30 b 4. If b is the A.M. betwee & b the fid the vlue of 5. The H.M of two umbers is 4 d their A.M. (A) d G.M. (G) stisfy the reltio A + G 7. Fid the umbers. 6. Fid the sum of first 0 terms of the series : Fid the sum of first 0 terms of the series : ( ) + (5 4 ) + (7 6 ) If 9 rithmetic mes d 9 hrmoic mes be iserted betwee d, prove tht 6 A + 5, where A is y rithmetic me d H the correspodig hrmoic me. H 9. Fid three umbers, b, c betwee d 8 such tht : (i) their sum is 5 (ii) the umbers,, b re cosecutive terms of A.P. d (iii) the umbers b, c, 8 re cosecutive terms of G.P. 0. If A, A ; G, G ; d H, H be two A.M. s, G.M. s d H.M. s betwee two umbers, the prove tht: G H G H A A H H.

31 ADVANCED LEVEL ASSIGNMENT. 5 7 Fid the sum of the series to ifiity.... Fid the th term d sum upto terms for the series: Fid T for the series If log ( + b) + log (c + d) 4. The fid miimum vlue of + b + c + d. 5. The turl umbers re grouped s (), (,, 4), (5, 6, 7, 8),... fid first elemet of th group Fid the sum upto ifiite terms of the series If ( )( )( ) tr, where t 8 r deotes the r th term of the series the fid r lim. r t r 8. Prove tht ( + ) ( + ) ( + 4 )... ( + ) ( ) 9. If cos, b si d c si cos, the show tht b c + c, 0,. b 0. If, b, c re i A.P.,, b, c re i H.P., the prove tht either b c or, b, c/ form G.P.. Let, b, be positive rel umbers. If, A, A, b re i A.P., G, G, b re i G.P. d, H, H, b re i H.P., show tht GG H H A A H H ( b) ( b) 9b. Let,... be positive rel umber i G.P. for ech. Let A,G, H be respectively the rithmetic me, geometric me, hrmoic me of,.... Fid expressio for the geometric me of G, G...G. I the term of A, A...A d H, H...H.. If, b, c re positive rel umbers, the prove tht [( + )( + b)( + c)] 7 > b 4 c 4.

32 4. The fourth power of commo differece of rithmetic progressio with itegrl etries is dded to the product of y four cosecutive terms of it prove tht the resultig sum is the squre of iteger. 5. Three cosecutive digits of three digit umber re i G.P. If the middle digit be icresed by, the they form A.P. If 79 subtrcted from this the we get the umber costitutig of sme three digits but i reverse order. The prove tht umber is divisible by 9.

33 ANSWERS Objective Questios () (). () () () 8. () 9. () 0.. (,b,c,d). (, b). (,b) 4. (,b,c) 5. (,b) 6. (,b) 7. (,b,c,d) 8. (,c) 9. (,b,c) 40. (,b,d) Miscelleous Assigmet. ().. () () 6. () 7. () () 0. A-(p); B-(r); C-(q); D-(p). A-(q); B-(r); C-(p); D-(r). (). (6) 4. (6) 5. () 6. () 7. (9) 8. (8) 9. () 0. (7). () Previous Yer Questios IIT-JEE/JEE-ADVANCE QUESTIONS. ().. (b,d) () () (8) 5. (9) () 9. (, d) 0. (5). (4) DCE QUESTIONS () 7. () () ()

34 .. () () MAINS QUESTIONS. ().. () () 0. (). () () () Bsic Level Assigmet () is series 4. (, 4, 6) or (6, 4, ) () : 8 6 ; [ ] c 6, b 4, d 9; b, c 6, d 8. () (4 ) ( )( )( ) , , b 8, c. Advced Level Assigmet. 6. T. T ( ) ( )( )