A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO 322, SECTOR 40 D, CHANDIGARH. Type (II) : Short Answer Type Questions : = 1, t 2

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1 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Type (I) : Vey Shot Aswe Type Questios : [0 Mk Ech]. Fid the fist five tems of the sequece fo which t, t d t + t + t +. How my tems e thee i the A.P. 0, 5, Is 55 tem of the sequece,, 5, 7,...? If yes fid which tem it is. 4. Fid the sum of the seies to 0 tems 5. If sum to tems of sequece be +, the fid fist tem, commo diffeece d lso fid sequece 6. A boy gees to wok t the te of oe upee the fist dy, two upees the secod dy, fou upees the thid dy, eight upees the fouth dy d so o i the moth of Apil. How much would he get o the 0th of Apil. 7. How my tems e thee i the G.P. 5, 0, 80,... 50? 8. The seveth tem of G.P. is 8 times the fouth tem. Fid the G.P. whe its 5th tem is How my tems of the seies must be tke to mke 80? 0. Fid + x x ( x) x. Fid to if x > 0. If 0 < <, the pove tht t + cot Type (II) : Shot Aswe Type Questios :. Fid the A.P. whose 7th d th tems e espectively 4 d Fid the sum of ll eve umbe betwee 0 to Iset 4 G.M's betwee 5 d Iset 7 A.M.'s betwee d 4 7. If m times the m th tem of A.P. is equl to times the th tem, fid its (m + )th tem. [0 Mks Ech] 8. Thee umbes e i A.P. thei sum is 7 d sum of thei sques is 75. Fid the umbes. 9. If, b, c e i A.P., pove tht b + c, c +, + b e lso i A.P. 0. If, b, c e i A.P., show tht (i) ( b) ( c) (b c) (ii) ( c) 4(b c). Fid the sum of the seies ( + b) + ( + b ) + ( b) to tems.. Fid the lest vlue of such tht to tems 500 SEQUENCE & SERIES # 7 A ONE INSTITUTE OF COMPETITIONS, PH ,

2 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH. Fid the sum to tems of the seies Expess s tiol umbe 5. Pove tht i ifiite G.P. whose commo tio is umeiclly less th oe, the tio of y tems to the sum of ll the succeedig tems is 6. Fid ( 7. ) to tems Type (III) : Log Aswe Type Questios: [04 Mk Ech] 8. If thee e ( + ) tems i A.P., the pove tht the tio of the sum of odd tems d thesum of eve tems is + : 9. If, b, c e i A.P. d x, y, z e i G.P. show tht x b c. y c. z b 0. If the A.M. d G.M. betwee two umbe be 5 d espectively, fid the umbes.. Pove tht the umbe... (9 digits) is ot pime umbe.. Fid the sum of the tem of the seies whose th tem is The cotiued poduct of thee umbes i G.P. is 6 d the sum of the poduct of them i pis is 56; Fid the umbe 4. If () /x (b) /y (c) /z d, b, c e i G.P., the pove tht x, y, z e i A.P. 5. If, b, c, d fou distict positive qutities i G.P. the show tht + d > b + c 6. If the sum of fist 0 tems of A.P. is 40 d the sum of fist 6 tems is 0, fid the sum of tems 7. The fist d lst tem of A.P. e d espectively. If S be the sum of ll the tem of the A.P., show tht the commo diffeece is S ( ) 8. The sum of fou iteges i A.P. is 4 d thei poduct is 945. Fid the umbes. 9. I set of fou umbes, the fist thee e i G.P. d the lst thee e i A.P. with commo diffeece of 6. If the fist umbe is sme s the fouth fid the fou umbe. Type (IV) : Vey Log Aswe Type Questios: [06 Mk Ech] 40. If S, S, S,... S P be the sum of tems of Aithmetic pogessios whose fist tems e espectively. p,,... d commo diffeece e,,...pove tht S + S + S +... S p ( + ) (p + ) 4 4. A umbe cosists of thee digits i G.P. the sum of the ight hd d left hd digit exceeds twice the middle digit by d the sum of the left hd d middle digits is two thid of the sum of the middle d ight hd digits. Fid the umbe SEQUENCE & SERIES # 8 A ONE INSTITUTE OF COMPETITIONS, PH ,

3 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 4. Fid 0th tem of the seies ? 4. If be oe A.M. d g d g be two geometic mes betwee b d c, pove tht g g bc 44. If x + y + z d x, y, z e positive umbes show tht ( x) ( y) ( z) 8xyz 45. If + b + c d > 0, b > 0, c > 0, fid the getest vlue of b c. 46. Fou diffeet iteges fom icesig A.P. oe of these umbe is equl to the sum of the sques of the othe thee umbes fid the umbes. 47. Fid the sum of tems of the seies How my tems e ideticl i the two ithmetic pogessios, 4, 6, 8,... Up to 00 tem d, 6, 9,... up to 80 tems. 49. The p th tem of A.P. is d qth tem is b, pove tht sum of its (p + q) tems is p q b b p q 50. A m ges to py off debt of Rs. 600 by 40 ul istllmet which e i A.P., whe 0 of the istllmets e pid he dies levig oe thid of the debt upid fid the vlue of the 8th istllmet. 5. Pove tht the sum of ltte hlf of tems of seies i A.P. is equl to oe thid of the sum of the fist tems. Sectio : Aithmetic Pogessio A-. A-. PART - I : SUBJECTIVE QUESTIONS I A.P. the thid tem is fou times the fist tem, d the sixth tem is 7 ; fid the seies. The thid tem of A.P. is 8, d the seveth tem is 0 ; fid the sum of 7 tems. A-. How my tems of the seies 9, 6,,... must be tke tht the sum my be 66? A-4. A-5. Fid the umbe of iteges betwee 00 & 000 tht e (i) divisible by 7 (ii) ot divisible by 7 Fid the sum of ll those iteges betwee 00 d 800 ech of which o divisio by 6 leves the emide 7. A-6. Fid the sum of 5 tems of the seies whose p th tem is 7 p +. A-7. The sum of thee umbes i A.P. is 7, d thei poduct is 504, fid them. A-8. If, b, c e i A.P., the show tht: (i) (b + c), b (c + ), c ( + b) e lso i A.P. (ii) b + c, c + b, + b c e i A.P. A-9. The fouth powe of the commo diffeece of ithmetic pogessio with itege eties is dded to the poduct of y fou cosecutive tems of it. Pove tht the esultig sum is the sque of itege. SEQUENCE & SERIES # 9 A ONE INSTITUTE OF COMPETITIONS, PH ,

4 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Sectio : Geometic Pogessio B-. The thid tem of G.P. is the sque of the fist tem. If the secod tem is 8, fid its sixth tem. B-. The cotiued poduct of thee umbes i G.P. is 6, d the sum of the poducts of them i pis is 56; fid the umbes B-. If the p th, q th, th tems of G.P. be, b, c espectively, pove tht q b p c p q. B-4. The sum of ifiite umbe of tems of G.P. is 4 d the sum of thei cubes is 9. Fid the seies. B-5. If, b, c, d e i G.P., pove tht : (i) (ii) ( b ), (b c ), (c d ) e i G.P.,, e i G.P. b b c c d Sectio : Hmoic d Aithmetic Geometic Pogessio C-. Fid the 4 th tem of H.P. whose 7 th tem is 0 d th tem is 8. C-. Sum the followig seies 4 (i) to tems. 7 5 (ii) to ifiity C-. Fid the sum of tems of the seies the th tem of which is ( + ). Sectio (D) : Mes, Iequlities A.M. G.M. H.M D-. D-. The ithmetic me of two umbes is 6 d thei geometic me G d hmoic me H stisfy the eltio G + H 48. Fid the two umbes. If betwee y two qutities thee be iseted two ithmetic mes A, A ; two geometic mes G, G ; d two hmoic mes H, H the pove tht G G : H H A + A : H + H. x D-. If x > 0, the fid getest vlue of the expessio 00. x x x... x D-4. Usig the eltio A.M. G.M. pove tht (i) (ii) (iii) t + cot ; if 0 < < (x y + y z + z x) (xy + yz + zx ) 9x y z. (x, y, z e positive el umbe) ( + b). (b + c). (c + ) bc ; if, b, c e positive el umbes Sectio (E) : Method of diffeece, t v v E-. Fid the sum to -tems of the sequece. (i) up to tems (ii) up to tems 00 SEQUENCE & SERIES # 0 A ONE INSTITUTE OF COMPETITIONS, PH ,

5 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH E-. Fid the sum to -tems of the sequece. (i) Sectio (F) : Miscelleous F-. F-. F (ii) The sum of thee umbes which e cosecutive tems of A.P. is. If the secod umbe is educed by & the thid is icesed by, we obti thee cosecutive tems of G.P., fid the umbes. If the p th, q th & th tems of AP e i GP. Fid the commo tio of the GP. Fid the sum of the tems of the seies whose th tem is (i) ( + ) (ii) PART - II : OBJECTIVE QUESTIONS * Mked Questios my hve moe th oe coect optio. Sectio : Aithmetic Pogessio A-. A-. A-. A-4. The fist tem of A.P. of cosecutive itege is p +. The sum of (p + ) tems of this seies c be expessed s (p + ) (p + ) (p + ) (p + ) (D) p + (p + ) If,,,... e i A.P. such tht , the is equl to (D) 900 If the sum of the fist tems of the A.P., 5, 8,..., is equl to the sum of the fist tems of the A.P. 57, 59, 6,..., the equls 0 (D) The sum of iteges fom to 00 tht e divisible by o 5 is (D) oe of these A-5. The iteio gles of polygo e i A.P. If the smllest gle is 0º & the commo diffeece is 5º, the the umbe of sides i the polygo is: (D) oe of these A-6. A-7. A-8. A-9. Coside A.P. with fist tem '' d the commo diffeece 'd'. Let S k deote the sum of its fist K Skx tems. If is idepedet of x, the Sx d/ d d (D) oe of these If x R, the umbes 5 +x + 5 x, /, 5 x + 5 x fom A.P. the '' must lie i the itevl: [, 5] [, 5] [5, ] (D) [, ) Thee e A.M's betwee d 54, such tht the 8th me: ( ) th me:: : 5. The vlue of is. 6 8 (D) 0 The sum of the seies log 4 log 4 log 4 log 4 ( + ) ( + ) ( + ) ( ) 8 (D) 4 ( + ) 4 is SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

6 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH A-0*. If,,..., e distict tems of A.P., the (D) A-*. If x, x +, x e thee tems of A.P., the its sum upto 0 tems is (D) 70 Sectio : Geometic Pogessio B-. B-. B-. B-4. B-5. B-6. B-7. The thid tem of G.P is 4. The poduct of the fist five tems is (D) oe of these If S is the sum to ifiity of G.P. whose fist tem is, the the sum of the fist tems is S S S S S (D) oe of these Coside ifiite geometic seies with fist tem '' d commo tio. If the sum is 4 d the secod tem is /4, the: 7 4, 7, 8 Fo sequece { }, d, 0. The is (D), 4 0 [4 + 9 ] 0 ( 0 ) (D) oe of these Suppose, b, c e i A.P. d, b, c e i G.P. if < b < c d + b + c, the the vlue of is (D), be the oots of the equtio x x + 0 d, the oots of x x + b 0 d umbes,,, (i this ode) fom icesig G.P., the, b, b, b (D) 4, b 6 The tiol umbe, which equls the umbe. 57 with ecuig deciml is B-8*. If sum of the ifiite G.P., p,,, p,,... is, the vlue of p is p p B-9*. Idicte the coect ltetive(s), fo 0 < < /, if: x 0 cos, y 0 si, z 0 (D) oe of these cos si, the: (D) xyz xz + y xyz xy + z xyz x + y + z (D) xyz yz + x SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

7 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Sectio : Hmoic d Aithmetic Geometic Pogessio C-. C-. Let the positive umbes, b, c, d be i A.P. The bc, bd, cd, bcd e: ot i A.P./G.P./H.P. i A.P. i G.P. (D) i H.P. If the sum of the oots of the qudtic equtio, x + bx + c 0 is equl to sum of the sques of thei ecipocls, the c, b, c e i b A.P. G.P. H.P. (D) oe of these C-. If + 4 ( + d) + ( + d) upto 8, the the vlue of d is: (D) oe of these Sectio (D) : Mes, Iequlities A.M. G.M. H.M D-. D-*. D-. D-4. If A, G & H e espectively the A.M., G.M. & H.M. of thee positive umbes, b, & c, the the equtio whose oots e, b, & c is give by: x Ax + G x G 0 x Ax + (G /H)x G 0 x + Ax + (G /H) x G 0 (D) x Ax (G /H) x + G 0 If the ithmetic me of two positive umbes & b ( > b) is twice thei geometic me, the : b is: + : : : 7 4 (D) : If, b, c, d e positive el umbes such tht + b + c + d, the M ( + b) (c + d) stisfies the eltio: 0 M M M (D) M 4 If,,,..., e positive el umbes whose poduct is fixed umbe c, the the miimum vlue of is (c) / ( + ) c / c / (D) ( + )(c) / Sectio (E) : Method of diffeece, t v v E-*. If ( ) ( + ) 4 + b + c + d + e, the + c b + d e 0, b /, c e i A.P. (D) c/ is itege Sectio (F) : Miscelleous F-. F-. F-. F-4. Suppose, b, c e i A.P. &, b, c <. If x to ; y + b + b +... to & z + c + c +... to, the x, y, z e i: A.P. G.P. H.P. (D) oe The sum of the fist -tems of the seies is eve. Whe is odd, the sum is ( ) 4 ( ) 4 ( ) (D) ( ) 4 ( ) If x b y c z d t d, b, c, d e i G.P., the x, y, z, t e i A.P. G.P. H.P. (D) oe of these The sum is equl to: /4 4/ (D) oe of these, whe is SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

8 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH PART - III : ASSERTION / REASONING. STATEMENT- : The seies fo which sum to tems, S, is give by S is A.P. STATEMENT- : The sum to tems of A.P. hvig o zeo commo diffeece is qudtic i, i.e., + b. STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is ot coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is flse (D) STATEMENT- is flse, STATEMENT- is tue (E) Both STATEMENTS e flse. STATEMENT- :,, 4, 8,... is G.P., 4, 8, 6, is G.P. d + 4, + 8, 4 + 6, 8 +,... is lso G.P. STATEMENT- : Let geel tem of G.P. (with positive tems) with commo tio be T k + d geel tem of othe G.P. (with positive tems) with commo tio be T k +, the the seies whose geel tem T k + T k + + T k + is lso G.P. with commo tio. STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is ot coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is flse (D) STATEMENT- is flse, STATEMENT- is tue (E) Both STATEMENTS e flse. STATEMENT- : The sum of the fist 0 tems of the sequece,,4,7,,6,,... is 450. STATEMENT- : If the successive diffeeces of the tems of sequece fom A.P., the geel tem of sequece is of the fom + b + c. STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is ot coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is flse (D) STATEMENT- is flse, STATEMENT- is tue (E) Both STATEMENTS e flse 4. STATEMENT- :,6, e i G.P., the 9,,8 e i H.P. STATEMENT- : If thee cosecutive tems of G.P. e positive d if middle tem is dded i these tems, the esultt will be i H.P. STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is ot coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is flse (D) STATEMENT- is flse, STATEMENT- is tue (E) Both STATEMENTS e flse 5. STATEMENT- : Miimum vlue of STATEMENT- : The lest vlue of si + b cos is (D) (E) si x cos x si x six fo x (si x )cos x b 0, is STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is tue d STATEMENT- is ot coect expltio fo STATEMENT- STATEMENT- is tue, STATEMENT- is flse STATEMENT- is flse, STATEMENT- is tue Both STATEMENTS e flse SEQUENCE & SERIES # 4 A ONE INSTITUTE OF COMPETITIONS, PH ,

9 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH PART - I : SUBJECTIVE QUESTIONS. Fid the sum i the th goup of sequece, (i) (), (, ); (4, 5, 6, 7); (8, 9,..., 5);... (ii) (), (,, 4), (5, 6, 7, 8, 9),.... Show tht,, 5 cot be the tems of sigle A.P.. If the sum of the fist m tems of A.P. is equl to the sum of eithe the ext tems o the ext p tems, the pove tht (m + ) (m + p). m p m 4. Fid the sum of the seies () () () up to + 5. If 0 < x < d the expessio exp {( + cos x + cos x + cos x + cos 4 x +... upto ) log e 4} stisfies the qudtic equtio y 0y , the fid the vlue of x. 6. I cicle of dius R sque is iscibed, the cicle is iscibed i the sque, ew sque i the cicle d so o fo times. Fid the limit of the sum of es of ll the cicles d the limit of the sum of es of ll the sques s. 7. Give tht e oots of the equtio Ax 4 x + 0 d, the oots of the equtio B x 6 x + 0, fid vlues of A d B, such tht,, & e i H.P. be 8. (i) If y be (ii) If b y c b ce y y, b ce c c de y y, c de b, the show tht,b,c,d e i G.P. e i A.P., the show tht 9 x+, 9 bx+,9 cx+, x 0 e i G.P. 9. If, b, c e positive el umbes, the pove tht b c + c + b bc ( + b + c). 0. If, b, c e positive el umbes d sides of the tigle, the pove tht ( + b + c) 7 ( + b c) (c + b) (b + c ). Sum the followig seies to tems. (i) ( + ) ( + ) ( + ). Sum of the followig seies (i) (ii) (ii) ( ) ( ) SEQUENCE & SERIES # 5 A ONE INSTITUTE OF COMPETITIONS, PH ,

10 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH. The sum of the fist te tems of AP is 55 & the sum of fist two tems of GP is 9. The fist tem of the AP is equl to the commo tio of the GP & the fist tem of the GP is equl to the commo diffeece of the AP. Fid the two pogessios. 4. Let,,..., be positive el umbes i geometic pogessio. Fo ech, let A, G, H be espectively the ithmetic me, geometic me & hmoic me of,,...,. Fid expessio fo the geometic me of G, G,..., G i tems of A, A,..., A, H, H,..., H. 5. Let, b be positive el umbes. If, A, A, b e i ithmetic pogessio,, G, G, b e i geometic pogessio d, H, H, b e i hmoic pogessio, show tht G G H H Sigle choice type A A H H ( b ) ( b ). 9 b PART - II : OBJECTIVE QUESTIONS. If x i > 0, i,,..., 50 d x + x x 50 50, the the miimum vlue of + x x equls to 50 x 50 (50) (50) (D) (50) 4. If,,,,...,, b e i A.P. d, g, g, g,...g, b e i G.P. d h is the hmoic me of d b, the h g g + g g g g is equl to h h (D) h. Oe side of equiltel tigle is 4 cm. The midpoits of its sides e joied to fom othe tigle whose mid poits e i tu joied to fom still othe tigle. This pocess cotiues idefiitely. The the sum of the peimetes of ll the tigles is 44 cm cm 88 cm (D) oe of these 4. If the sum of tems of G.P. (with commo tio ) begiig with the p th tem is k times the sum of equl umbe of tems of the sme seies begiig with the q th tem, the the vlue of k is: p/q q/p p q (D) p + q 5. If P, Q be the A.M., G.M. espectively betwee y two tiol umbes d b, the P Q is equl to b b 6. I G.P. of positive tems, y tem is equl to the sum of the ext two tems. The commo tio of the G.P. is cos 8 si 8 cos 8 (D) si 8 b b 7. If upto 6, the / /4 /8 (D) oe of these 8. If,,... e i A.P. with commo diffeece d 0, the the sum of the seies (si d) [cosec cosec + cosec cosec cosec cosec ] (D) b sec sec cot cot cosec cosec (D) t t SEQUENCE & SERIES # 6 A ONE INSTITUTE OF COMPETITIONS, PH ,

11 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 9. Sum of the seies S is (D) oe of these 5 0. If H , the vlue of is H + H H (D) H +. If S, S, S e the sums of fist tul umbes, thei sques, thei cubes espectively, the S( 8S) is equl to S 9 (D) 0. Coside the followig sttemets : S : Equl umbes e lwys i A.P., G.P. d H.P. S : If x > d,, x x b c x S : If, b, c be i H.P., the b, b, c b S 4 : e i G.P., the, b, c e i A.P. will be i AP If G d G e two geometic mes d A is the ithmetic me iseted betwee two positive umbes, the the vlue of G G is A. G G Stte, i ode, whethe S, S, S, S 4 e tue o flse FTFT TTTT FFFF (D) TFTF Moe th oe choice type. The sides of ight tigle fom G.P. The tget of the smllest gle is If b, b, b (b i > 0) e thee successive tems of G.P. with commo tio, the vlue of fo which the iequlity b > 4b b holds is give by > 0 < <.5 (D) 5. PART - III : MATCH THE COLUMN (D) 5. Colum Colum If log 5, log 5 ( x 5) d log 5 ( x 7/) e i A.P., (p) 6 the vlue of x is equl to Let S deote sum of fist tems of A.P. If S S, (q) 9 the S S is 8 6 Sum of ifiite seies is () (D) The legth,bedth, height of ectgul box e i G.P. The (s) volume is 7, the totl sufce e is 78. The the legth is SEQUENCE & SERIES # 7 A ONE INSTITUTE OF COMPETITIONS, PH ,

12 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH. Colum Colum If log x y, log z x, log y z e i G.P., xyz 64 d x,y,z (p) x e i A.P., the is equl to y The vlue of is equl to (q) If x, y, z e i A.P., the () (x + y z) (y + z x) (z + x y) kxyz, whee k N, the k is equl to (D) Thee e m A.M. betwee d. If the tio of the (s) 4 Compehesio # 7 th d (m ) th mes is 5 : 9, the 7 m is equl to PART - IV : COMPREHENSIONS ( ) We kow tht f(), ( )( ) g(), ( ) h(). g() g( ) must be equl to ( ) (D). Getest eve tul umbe which divides g() f(), fo evey, is 4 6 (D) oe of these. f() + g() + h() is divisible by oly if oly if is odd oly if is eve (D) fo ll N Compehesio # I sequece of (4 + ) tems the fist ( + ) tems e i AP whose commo diffeece is, d the lst ( + ) tems e i GP whose commo tio 0.5. If the middle tems of the AP d GP e equl, the 4. Middle tem of the sequece is. 5. Fist tem of the sequece is (D) Noe of these 6. Middle tem of the GP is (D).. (D) SEQUENCE & SERIES # 8 A ONE INSTITUTE OF COMPETITIONS, PH ,

13 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH PART - I : IIT-JEE PROBLEMS (PREVIOUS YEARS) * Mked Questios my hve moe th oe coect optio.. If 0,, the t x x + x x is lwys gete th o equl to: [IIT-JEE-00, Sc., (, ), 84] t (D) sec. If, b & c e i ithmetic pogessio d, b & c e i hmoic pogessio, the pove tht eithe b c o, b & c e i geometic pogessio. [IIT-JEE-00, Mi, (4, 0), 60]. A ifiite G.P. hs fist tem s x d sum upto ifiity s 5. The the ge of vlues of x is: [IIT-JEE-004, Sc., (, ), 84] x 0 x 0 0 < x < 0 (D) 0 x 0 4. I the qudtic equtio x + bx + c 0, 0, b 4c d +, +, + e i G.P. whee e the oot of x + bx + c 0, the [IIT-JEE-005, Sc., (, ), 84] 0 b 0 c 0 (D) 0 5. If totl umbe of us scoed i mtches is ( 4 + ) whee > d the us scoed i the k th mtch e give by k. + k, whee k, fid [IIT-JEE-005,Mi, (, 0), 60] 6. If ( ) 4 d b, the fid the miimum tul umbe 0 such tht b > > 0 [IIT-JEE 006, (6, 0), 84] Compehesio # Let V deotes the sum of the fist tems of ithmetic pogessio (A.P.) whose fist tem is d the commo diffeece is ( ). Let T V + V d Q T + T fo,,... [IIT-JEE 007, Ppe-, (4, ), 8] 7. The sum V + V V is ( + ) ( + ) ( + ) ( + + ) ( + ) (D) ( + ) 8. T is lwys odd umbe eve umbe pime umbe (D) composite umbe 9. Which oe of the followig is coect sttemet? Q, Q, Q,... e i A.P. with commo diffeece 5 Q, Q, Q,... e i A.P. with commo diffeece 6 Q, Q, Q,... e i A.P. with commo diffeece (D) Q Q Q... SEQUENCE & SERIES # 9 A ONE INSTITUTE OF COMPETITIONS, PH ,

14 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Compehesio # Let A, G, H deote the ithmetic, geometic d hmoic mes, espectively, of two distict positive umbes. Fo, let A d H hve ithmetic, geometic d hmoic mes s A, G, H espectively. [IIT-JEE 007, Ppe-, (4, ), 8] 0. Which oe of the followig sttemets is coect? G > G > G >... G < G < G <... G G G... (D) G < G < G 5 <... d G > G 4 > G 6 >.... Which oe of the followig sttemets is coect? A > A > A >... A < A < A <... A > A > A 5 >... d A < A 4 < A 6 <... (D) A < A < A 5 <... d A > A 4 > A 6 >.... Which oe of the followig sttemets is coect? H > H > H >... H < H < H <... H > H > H 5 >... d H < H 4 < H 6 <... (D) H < H < H 5 <... d H > H 4 > H 6 >.... Suppose fou distict positive umbes,,, 4 e i G.P. Let b, b b +, b b + d b 4 b + 4 STATEMENT - : The umbes b, b, b, b 4 e eithe i A.P. o i G.P. STATEMENT- : The umbes b, b, b, b 4 e i H.P. STATEMENT- is Tue, STATEMENT- is Tue ; STATEMENT- is coect expltio fo STATEMENT- STATEMENT- is Tue, STATEMENT- is Tue ; STATEMENT- is NOT coect expltio fo STATEMENT- STATEMENT- is Tue, STATEMENT- is Flse (D) STATEMENT- is Flse, STATEMENT- is Tue [IIT-JEE 008, Ppe-, (, ), 8] 4. If the sum of fist tems of A.P. is c, the the sum of sques of these tems is [IIT-JEE - 009, Ppe-, (, ), 80] (4 ) c 6 (4 ) c 6 (4 ) c (D) (4 ) c 6 (m ) m 5*. Fo 0 < <, the solutio(s) of cos ec cos ec 4 is(e) m [IIT-JEE - 009, Ppe-, (4, ), 80] 5 (D) 6. Let S k, k,,..., 00, deote the sum of the ifiite geometic seies whose fist tem is 00 the commo tio is. The the vlue of + k ( k k )S k is 00! 00 k [IIT-JEE - 00, Ppe-, (, 0), 84] k d k! 7. Let,,,..., be el umbes stisfyig 5, 7 > 0 d k k k fo k, 4,...,. If , the the vlue of is equl to [IIT-JEE - 00, Ppe-, (, 0), 79] 8. Let,,,..., 00 be ithmetic pogessio with d S p i, p 00. Fo y itege with 0, let m 5. If S S m p i does ot deped o, the is [IIT-JEE 0, Ppe-, (4, 0), 80] SEQUENCE & SERIES # 0 A ONE INSTITUTE OF COMPETITIONS, PH ,

15 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 9. The miimum vlue of the sum of el umbes 5, 4,,, 8 d 0 whee > 0 is [IIT-JEE 0, Ppe-, (4, 0), 80] 0. Let,,,... be i hmoic pogessio with 5 d 0 5. The lest positive itege fo which < 0 is [IIT-JEE 0, Ppe-, (, ), 66] 4 (D) 5 PART - II : AIEEE PROBLEMS (PREVIOUS YEARS). If x, x, x d y,y, y e both i GP with the sme commo tio, the the poits (x, y ), (x,y ) d (x,y ): () lie o stight lie () lie o elipse [AIEEE 00] () lie o cicle (4) e vetices of tigle.. Let T be the th tem of AP whose fist tem is d commo diffeece is d. If fo some positive iteges m &, m, T m d T m, the d equls : [AIEEE 004] () 0 () () m. If x, y b, z (4) m c whee,b,c e i AP d <, b <, c <, the x,y,z e i : () HP () Aithmetico Geometic Pogessio [AIEEE 005] () AP (4) GP 4. If i ABC, the ltiudes fom the vetices A, B, C o opposite sides e i H.P., the si A, si B, si C e i- () G.P. () A.P. [AIEEE 005] () Aithmetico-Geometic pogessio (4) H.P. 5. Let,,,... be tems of AP. If p q p 6, p q, the equls : [AIEEE 006] q 7 4 () () () (4) If,,..., e i HP, the the expessio is equl to : [AIEEE 006] () ( ) ( ) () () ( ) (4) ( ) 7. I geometic pogessio cosistig of positive tems, ech tem equls the sum of the ext two tems. The the commo tio of this pogessio equls [AIEEE 007] () ( 5) () 5 () 5 (4) ( 5 ) 8. A peso is to cout 4500 cuecy otes. Let deote the umbe of otes he couts i the th miute. If d 0,,...e i AP with commo diffeece, the the time tke by him to cout ll otes is [AIEEE 00] () 4 miutes () 5 miutes () 5 miutes (4) 4 miutes 9. A m sves Rs. 00 i ech of the fist thee moths of his sevice. I ech of the subsequet moths his svig iceses by Rs. 40 moe th the svig of immeditely pevious moth. His totl svig fom the stt of sevice will be Rs. 040 fte : [AIEEE 0] () 8 moths () 9 moths () 0 moths (4) moths Let be the th tem of A.P. If () () d, the the commo diffeece of the A.P. is : () (4) 00 [AIEEE 0]. The sum of fist 0 tems of the sequece 0.7, 0.77, 0.777,..., is [AIEEE - 0, (4, ¼),60] 7 () ( ) () ( ) () ( ) (4) ( ) SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

16 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH BOARD LEVEL SOLUTIONS. Give, t, t, t + t + t + Puttig, we get t t + t +, we get t 4 t + t + 5, we get t 5 t + t Thus the fist five tems of the give sequece e,,, 5 d 8.. Let the umbe of tems be give t 00, 0, d 5, we hve to fid. Now t + ( )d ( )5 o 80 ( ) 5 o 6 7. If possible let th tem of the sequece be 55. Now t + ( )d Hee t 55,, d 55 + ( ) o 56 8 Hece 55 is 8th tem of the give sequece Note : If does ot come out to be itege, the 55 will ot be tem of the give sequece. 4. Tems of the give seies e i A.P. whose commo diffeece d 4 d fist tem 99 Now sum of 0 tems of the seies 0 S 0 [.99 + (0 ) ( 4)] 0 (98 76) 0 5. Give sum of tem of y sequece + we kow S t + t + t +... t + Put, S t + Put, S t + t () + () 8 Put, S t + t + t + 5 S S t 5 S S t 7 Hece sequece is, 5, 7,... which is A.P. whose fist tem is d commo diffeece is. Note : If geel tem of y sequece is lie expessio of. (t + b) d sum of tems is qudtic expessio (S + b + c) the sequece is A.P. 6. Hee,, 0, to fid t Now t.() 0 9 Hece the boy will get 9 upees o 0th of Apil 7. Let the umbe of tems be Give, 5, 4, t 50 t o Give t 7 8t whee d e the fist tem d commo tio espectively of the G.P. o 8 Also t o () 4 48 o 6 48 Hece equied G.P. is, 6,, 4, Let the sum of tems of the give seies be 80 ( ) S o 80 o 6560 o ( ) Hee tems of give seies e i G.P. d x, Also x Now S x x x x + x <. Let S ( ) ( + + ) 0(0 ) ( ) If 0 < <, the t, cot both e positive umbe A.M. G. M. t cot (t. cot) / t + cot. th tems of A.P. whose fist tems is d commo diffeece is d is give by t + ( )d Give whe 7, t d...(i) whe, t d...(ii) subtctig (i) fom (ii) we get 0 6d d 5 Puttig d 5 i (i), we get Hece the equied A.P. is 4, 9, 4, 9, 4, Fist eve umbe betwee 0 d 999 is 0 d the lst eve umbe is 998 d diffeece betwee two cosecutive eve umbe is. Hece 0, d, t 998 t + ( ) d ( ) o ( ) 896 SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

17 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH o Now sum of ll eve umbes betwee 0 to 999 (Fist tem + Lst tem) 449 ( ) Let G, G, G, G 4 be the fou G.M's betwee 5 d 60 5, G, G, G, G 4, 60 will be i G.P. Now 60 6th tem of G.P ( 5) o 5 5 Now G 5 0 G 5 0 G 5 40 G Let A, A, A, A 4, A 5, A 6, A 7 be the seve A.M.'s betwee d 4, A, A, A, A 4, A 5, A 6, A 7, 4 wil be i A.P. Now 4 9th tem of A.P. + 8d + 8d [ ] o 8d d 4 Now A + d A + d A + d A 4 + 4d A 5 + 5d A 6 + 6d A 7 + 7d Let be the fist tem d d the commo diffeece of A.P. Give tht t mt m [ + ( )d] m[ + (m )d] o (m ) d[( ) m(m )] o (m ) d[(m ) (m )] o (m ) d(m ) [ (m +)] o d( m ) [ (m + )] [ m ] o d[m + ]...() Now (m + ) th tem t m+ + (m + )d 0 [fom (i)] 8. Let the thee umbes i A.P. be d,, + d Give ( d) + + ( + d) 7 o 7 9 d ( d) + + ( + d) 75 o + d d d + d 75 o + d 75 o (9) + d 75 o d 75 4 o d 6 d ± 4 If d 4, the thee umbe e 5, 9, 4 If d 4 the thee umbes e 4, 9, 5 9., b, c e i A.P. ( + b + c), b ( + b + c), c ( + b + c) e i A.P. (b + c), ( + c), ( + b) e i A.P. b + c, c +, + b e i A.P. 0. give, b, c e i A.P. Let d be commo diffeece The b + d, c + d Now (b c) ( + d d) d, c ( + d) d Hece ( b) c (b c) (ii) ( c) ( d) 4d 4(b c) 4[( + d) ( + d)] 4[ + d + d d] 4d Hece ( c) 4(b c). Tems of give seies e i A.P. Whose fist tem ( + b) d commo diffeece ( + b ) ( + b) b Now sum of tems of give seies S [( + b) + ( ) ( b)] [( + b ) + b ( )b] ( + b ) + b( ). Give to tems 500 o [() + ( ) ()] 500 o o 500 But is positive itege 500 o.6 lest vlue of.. Let S to tems 8[ to tems] 9 8 [ to tems] 9 8 [(0 ) + (0 ) + (0 ) +... tems] 9 8 [( ] ( tems) [ ] 8 4. Let x to to to to Let fist tems of ifiite G.P. is. the,,,... to SEQUENCE & SERIES # A ONE INSTITUTE OF COMPETITIONS, PH ,

18 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Give G.P.) t S S y x, z y x ( ) Now x b c y c. z b x b c (x) c (x ) b ( ) (x) d (x) d (x ) d x d+d d () d d x 0. x 0 ( ( )). ( ) 6. ( ) ( ) ( ) ( ) ( ) ( ) ( ) Let S... to tem tem ( times)... tem Let the A.P. be, + d, + d, + d,... + d sum of its odd tems + ( + d) + ( + 4d) +... to ( + ) tems [ + ( + ) d] ( + ) ( + d) sum of eve tems ( + d) + ( + d) +... to tems [( + d) + ( ) d] ( + d) sum of odd tems sum of eve tems 9. Give, b, c e i A.P. b c b commo diffeece (Let d) (c b) + (b ) d o c d d x, y, z e i G.P. y z (commo tio of x y 0. Let, b be the two umbes Give A.M. betwee d b 5 b 5 + b 0...(i) is G.M. betwee d b,, b will be i G.P. b b 9...(ii) Put vlue of b fom (ii) ito (i) , 9 Whe, b 9 Whe 9, b Thus the umbes e d 9 o 9 d. We hve... (9 digits) (0 ) ( ) ( ) Thus... (9 digits) is ot pime umbe. Let t S t ( )( ) ( ) ( + ) ( + ) ( + ) Let the thee umbes i G.P. is,, Give SEQUENCE & SERIES # 4 A ONE INSTITUTE OF COMPETITIONS, PH ,

19 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 6 8. Let the fou umbe e d, d, + d, + d Give, ( d) + ( d) + ( + d) + ( + d) 4 Also o d ( d) ( d) ( + d) ( + d) 945 o o ( 9d ) ( d ) o o (6 9d ) (6 d ) 945 o d 4 40 d o d 4 40d o o o d 4 d 9d o (d ) (d 9) 0 Sice umbe e iteges If, the the umbe e 8, 6, d 9 If, the the umbe e 8, 6, d d Hece Fou iteges e, 5, 7, 9 o 9, 7, 5, 4. Give /x b /y c /z (Let) the x, b y, c z 9. Let the lst thee umbes i A.P. be, + 6, +, b, c e i G.P., the d the fist umbe be. b c Hece the fou umbes e,, + 6, + y x. z Give ( + )...(i) y x+z d,, + 6 e i G.P. ( + 6) y x + z o ( + ) ( + 6) [ + ] Hece x, y, z e i A.P. o Fom (i) Give, b, c, d e i G.P. Hece the fou umbs e 8, 4, d 8 Fo fist thee tems A.M. > G.M. c > b...(i) b d Fo lst thee tems > c...(ii) b c d dd (i) + (ii), we get + d > b + c > b + c 6. Hee S 0 40, S 6 0, to fid S Now 0 40 S 0 [ + (0 ) d] o 40 5( + 9d) o + 9d 8...(i) 6 d 0 S 6 [ + (6 )d] o + 5d 40...(ii) subttig (ii) (i) 6d d Put d i (i) we get 5 Now S [ 5 + ( ) ] Let is fist tem d d is commo diffeece of A.P. The + ( )d...(i) whee is umbe of tems i A.P. S d S ( + )...(ii) S Put vlue of i (i) + d o d S S ( ) ( ) 40. S [() + ( ) ()]. ( ) S [() + ( ) ()]. ( ) S [() + ( ) ()] ( ) S P [P + ( )P] P Now S + S + S S P ( ) [ to P tems] ( ) P(P ) p ( + ) (P + ) 4 4. Let the thee digits be, d Give + + o ( +) o ( ) Also ccodig to questio, + ( + ) O ( + ) ( + ) o ( + ) ( ) 0, Whe itege, 4 which is ot possible, fo is Hece 4, 4. 6, equied umbe is SEQUENCE & SERIES # 5 A ONE INSTITUTE OF COMPETITIONS, PH ,

20 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 4. Let S t + t...(i) 46. Let the umbe be d,, + d, + d Agi S t + t...(ii) whee, d I, d > 0 Subtct (ii) fom (i) Give ( d) + + ( + d) ( + d) 0 + [ to ( ) tems] t o d d + ( ) 0 o t + [ + ( ) ] + ( ) + t 0 (0) b c 4. Give A.M. betwee b d c g d g e two G.M. betwee b d c b, g, g, c e i G.P. g b, g b, c b whee is commo tio of G.P. Now g g (b) + (b ) b ( + ) b. b c c c b b b c b c bc (b + c) bc() b b c c bc Thus g g bc 44. Sice A.M. G.M. y z yz...(i) z x zx...(ii) x y xy...(iii) Multiplyig (i), (ii) d (iii), we get ( x y)(y z)(z x) 8xyz 8 o ( x) ( y) ( z) 8xyz ( x + y + z ) 45. Tkig A.M. d G.M. of umbe b b b c c,,,,,,, we get A.M. G.M. b c... b c 7 / 7 b c 7 o b c 7.. o o b c Getest vlue of b c / 7 4 d 4 4()(. 6 o d Sice, d is positive itege + 6 > 0 o 6 < 0 o ) < o < < 6 6 sice is itege 0 the d [ ] o 0 sice d > 0 d Hece, the umbes e, 0,, 47. Let S t + t...(i) d S t + t + t...(ii) Subtctig (ii) fom (i), we get 0 + [ to ( ) tems] t o t + [ to ( ) tems] ( ) +. [.4 + ( ) ] + ( ) ( + ) + S t + ) + ( )( ) + 6 ( ) ( ) 48. Let tems be ideticl Now sequece of ideticl tems is 6,, 8 Its th tems 6 + ( ) th tem of the sequece, 4, 6, (00 ) () 00 d 80 th tem of the sequece, 6, 9,... + (80 ) () 40 Sice, lst tem i.e. th tem of the sequece of ideticl tems cot be gete th o o 6 Hece tems e ideticl 49. Let A be the fist tem d D is commo diffeece of A.P. SEQUENCE & SERIES # 6 A ONE INSTITUTE OF COMPETITIONS, PH ,

21 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Give t p A + (p ) D...(i) t q b A + (q ) D b...(ii) EXERCISE # Subtctig (i) (ii), we get (p q)d b D b p q Addig (i) d (ii) we get A + (p + q ) D + b A + (p + q ) D + b + D b A + (p + q ) D + b + [fom vlue of D] p q Now S p+q p q p q [A + ( + q )] b b p q 50. Let the Fist istllmet be d commo diffeece of A.P. be d. 40 Give 600 sum of 40 tems [ + (40 ) d] o ( + 9d) o + 9d 80...(i) Afte 0 istllmets oe thid of the debt is upid Hece is upid d 400 is pid 0 Now 400 ( + (0 )d) o d...(ii) Substctig (ii) fom (i), we get 0 0d d Fom (i), o Now vlue of the 8th istllmets + (8 )d Rs Rs Sum of ltte hlf of tems S S [ + ( )d] [ + ( )d] Whee is the fist tem d d the commo diffeece of A.P. [4 + ( )d ( )d] [ + (4 + )] [ + ( ) d]. [ + ( ) d] S sum of the fist tems Sectio : PART - I A-., 5, 8,... A-. 6 A-. A-4. 8, 77 A A A-7. 4, 9, 4 Sectio : B-. 8 B-., 6, 8 B-4. 6,, /,... Sectio : C-. C Sectio (D) : D-. 4, b 8 D-. Sectio (E) : C-. (i) 4 0 E-. (i) + 4 (ii) 7 ( ) E-. (i) (ii) Sectio (F) : 4( )( ) ( + ) ( + ) ( + ) ( + ) 0 F-., 7, o, 7, F-. F-. (i) 6 ( + ) ( + 7) (ii) ( + + ) + PART - II Sectio : A-. (D) A-. (D) A-. A-4. A-5. A-6. A-7. (D) A-8. A-9. (D) A-0*. (BD) A-*. (AB) q p q (ii) 8 SEQUENCE & SERIES # 7 A ONE INSTITUTE OF COMPETITIONS, PH ,

22 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Sectio : PART - II B-. B-. B-. (D) B-4. B-5. (D) B-6. B-7. B-8*. (AC) B-9*. (BC) Sectio : C-. (D) C-. C-. Sectio (D) : D-. D-*. (ABC) D-. D-4. Sectio (E) : E-*. (ABCD) Sectio (F) : F-. F-. F-. F-4. PART - III... (D) EXERCISE # PART - I. (i) ( + ) (ii) ( ) A ; B 8,, 6. R ; 4 R. (i) (/5) ( + ) ( + ) ( + ) ( + 4) (ii). (i) 5 54 ( ) 4 ( ) (ii) ( ) ; s ( ). ( ) ; (/ + 5/ + 65/6 +...) G.P ( ) ;.... A.P G (A K k Hk ) (D) 6. (D) (BC) 4. (ABCD) PART - III. (p), (p), (q), (D) (q). (), (p), (s), (D) (p) PART - IV... (D) (D) EXERCISE # PART - I miimum tul umbe (D) *.(CD) o 9, both d 9 (The commo diffeece of the ithmtic pogessio c be eithe 0 o 6, d ccodigly the secod tem c be eithe, o 9 ; thus the swes, o 9, o both d 9 e cceptble.) (D) PART - II. (). (). () 4. () 5. () 6. () 7. (4) 8. () 9. (4) 0. (). () SEQUENCE & SERIES # 8 A ONE INSTITUTE OF COMPETITIONS, PH ,

23 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Sigle choice type PART - I : OBJECTIVE QUESTIONS. Let { } d {b } e two sequeces give by ( x) + ( y) d b ( x) ( y) fo ll N. The vlue of... is equl to x y x y b. If (00) (4007) (4) d () (00) + () (00) + () (00) (00) () (00) (4) (x)., the x equls (D) 00 x + log 4 x + log 8 x + log 6 x , the x equls to 4 (D) 5. If x > 0, d log x + log 4. If t ( + ) ( + ), the the vlue of ( ) / x y b is t 4 5. If, b, c e i A.P., p, q, e i H.P. d p, bq, c e i G.P., the p + p c + c c c q b + b q / (D) (D) xy b / (D) q b p 6. The commo diffeece d of the A.P. i which T 7 9 d T T T 7 is lest, is / is equl to (D) oe of these 6 7. The H.M. betwee two umbes is, thei A.M. is A d G.M. is G. If A + G 6, the the umbes 5 e 6, 8 4, 8, 8 (D), 8 8. If,,... e fist tems;,, 5... e commo diffeeces d S, S, S... e sums of tems of give p AP s; the S + S + S S p is equl to p(p ) (p ) p(p ) 9. If d b e p th d q th tems of AP, the the sum of its (p + q) tems is p q b b p q p q b b p q (D) p(p ) p q b b p q (D) oe of these SEQUENCE & SERIES # 9 A ONE INSTITUTE OF COMPETITIONS, PH ,

24 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 0. If S t ( 9 ), the 6. t equls (D). The sum of those iteges fom to 00 which e ot divisible by o 5 is (D) 6. If, b, c e i GP, b, c, b c e i HP, the the vlue of + 4b + c is 0 (D) Moe th oe choice type. The vlue of x ( ) x x is x x x (D) x 4. Let, x, b be i A.P;, y, b be i G.P d, z, b be i H.P. If x y + d 5z, the y xz x > y > z 9, b (D) /4, b 9/4 5. If, log y x, log z y, 5 log x z e i A.P., the z x x y z y (D) x y z PART - II : SUBJECTIVE QUESTIONS. I A.P. of which is the Ist tem, if the sum of the Ist ' p ' tems is equl to zeo, show tht the sum of ( p the ext ' q ' tems is q ) q. p. The umbe of tems i A.P. is eve ; the sum of the odd tems is 4, sum of the eve tems is 0, d the lst tem exceeds the fist by 0½; fid the umbe of tems.. A m ges to py off debt of Rs. 600 by 40 ul istllmets which fom ithmetic seies. Whe 0 of the istllmets e pid he dies levig thid of the debt upid. Fid the vlue of the fist istllmet. 4. If the p th, q th d th tems of A.P. e, b, c espectively, show tht (q ) + ( p) b + (p q) c The sum of fist p-tems of A.P. is q d the sum of fist q tems is p, fid the sum of fist (p + q) tems. 6. If b is the hmoic me betwee d c, the pove tht + b b c x. c SEQUENCE & SERIES # 40 A ONE INSTITUTE OF COMPETITIONS, PH ,

25 A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 7. The vlue of x + y + z is 5 if, x, y, z, b e i AP while the vlue of (/x) + (/y) + (/z) is 5/ if, x, y, z, b e i HP. Fid d b. 8. Fid the vlue of S. 5 (5 )(5 ) d hece S. 9. If is oot of the equtio x² ( c) x (² + c²) ( + c) 0 d if HM s e iseted betwee d c, show tht the diffeece betwee the fist d the lst me is equl to c( c). 0. Cicles e iscibed i the cute gle so tht evey eighbouig cicles touch ech othe. If the dius of the fist cicle is R, the fid the sum of the dii of the fist cicles i tems of R d.. Let, b, c be positive el umbes, the pove tht (i) (ii) (iii) (iv) b c + b b b b + + c b c + b c b c + (b c) b c b + c + ( c) c c 9 b c + b + c +, if bc c ( b). Let A, G, H be A.M., G.M. d H.M. of thee positive el umbes, b, c espectively such tht G AH, the pove tht, b, c e tems of GP. PART - I (D).. (AC) 4. (ABC) 5. (ABCD) PART - II. 8 tems. Seies,, 4,.... Rs (p + q) 7., b 9 OR b, R si 0. si si si SEQUENCE & SERIES # 4 A ONE INSTITUTE OF COMPETITIONS, PH ,