jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt ( )( ) n n + 1 b c d e a a b c d e = + a + b c
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1 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF f/u fpkj Hkh# tu] ugh vkjehks dke] fif s[k NksMs qj e/;e eu dj ';kea iq#"k flg ldyi dj] lgs fif vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks VsdAA jfp% eku /kez iz.ksk l~xq# Jh j.knksmklth egkjkt ASSERTION & REASON FOR SEQUANCE AND SERIES Sme questis (Asseti Res type) e give belw. Ech questi ctis Sttemet (Asseti) d Sttemet (Res). Ech questi hs chices (A), (B), (C) d (D) ut f which ONLY ONE is cect. S select the cect chice: (A) Sttemet is Tue, Sttemet is Tue; Sttemet is cect explti f Sttemet. (B) Sttemet is Tue, Sttemet is Tue; Sttemet is NOT cect explti f Sttemet. (C) Sttemet is Tue, Sttemet is Flse. (D) Sttemet is Flse, Sttemet is Tue. 9. Sttemet :I the expessi (x ) (x )... (x 0), cefficiet f x 9 is equl t 7. ( ) Sttemet : =, N. = i 0. Let, b, c, d e fu psitive umbe b c d Sttemet : Sttemet : b c d e e. Let, b, c d d be distict psitive el umbes i H.P. Sttemet : d > b c Sttemet : b c d e. d e = d b c. Let, R {0,, } d be eve umbe. Sttemet :..... = ( ) /. Sttemet : Pduct f k th tem fm the begiig d fm the ed i G.P. is idepedet f k.. Sttemet : Let p, q, R d 7pq (p q ) d p q =, the p q is equl t. Sttemet : If A,G, d H e A.M., G.M., d H.M. f psitive umbes,,,..., the H G A.. Sttemet : The sum f seies. ( ) ( ) ( ) ( ).... ( ) is. 6 ( )( ) Sttemet : The sum f y seies S c be give s, S = T is the geel te f the = seies.. Sttemet : P is pit (, b, c). Let A, B, C be imges f P i yz, zx d xy ple espectively, the equti f ple must be x y z =. Sttemet : The diecti ti f the lie jiig igi d pit (x, y, z) must be x, y, z. 6. Sttemet : If A, B, C, D be the vetices f ectgle i de. The psiti vect f A, B, C, D be, b, c, d espectively, the. c = b.d. Sttemet : I tigle ABC, let O, G d H be the cicumcete, cetid d thcete f the tigle ABC, the OA OB OC = OH. ( ) b 7. Sttemet-: 7... upt tems = Sttemet-: is HM f & b if = - b 8. Sttemet-:... (up t 9 tems) is pime umbe Sttemet-: If b c, c b, b c e i A.P., the,, e ls i A.P. f 6
2 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF 9. Sttemet-: F ifiite G.P. whse fist tem is d cmm ti is, the S = whee Sttemet-: A, G, H e ithmetic me, Gemetic me d hmic me f tw psitive el umbes & b. The A, G, H e i G.P. 60. Sttemet-: (up t 9 tems) is pime umbe. Sttemet-: If b c c, b, b c Ae i A.P., the,, e ls i A.P. 6. Sttemet-: The sum f ll the pducts f the fist psitive iteges tke tw t time is ( ) ( ) ( ) Sttemet-: i j = (... ) ( ) i i< j 6. Sttemet-: Let the psitive umbes, b, c, d, e be i AP, the bcd, bce, bde, cde, bcde e i HP Sttemet-: If ech tem f A.P. is divided by the sme umbe k, the esultig sequece is ls 6. Sttemet-: If, b, c e i G.P.,,, e i H.P. lg lg b lg c Sttemet-: Whe we tke lgithm f the tems i G.P., they ccu i A.P. 6. Sttemet-: If p q = the p q hee p, q, R S-: If the qutities e psitive the weighted ithmetic me is gete th equl t gemetic me. 6. Sttemet-: S = / /... = / = S-: Sum f tems f G.P. with fist tem s d cmm ti s i give by, >. 66. Sttemet-: - / /... is gemetic sequece. Sttemet-: Tems f sequece e psitive umebs. 67. Sttemet-: The sum f the ifiite A.P..... is give by = = Sttemet-: The sum f ifiite G.P. is give by whee < is fist tem d is cmm ti. 68. Sttemet-: If,,,.. e psitive el umbes whse pduct is fixed umbe C, the the miimum vlue f.. is /. Sttemet-: If,,,.. R... /. the (... ) 69. Sttemet-: If (b c) x b (c ) x c( b) = 0 hs equl ts, the, b, c e i H.P. Sttemet-: Sum f the ts d pduct f the t e equl x 70. Sttemet-: lim = 0 f evey > 0! Sttemet-: Evey sequece whse th tem ctis! i the demit cveges t ze. 7. Sttemet-: Sum f ifiite gemetic seies with cmm ti me th e is t pssible t fid ut. S-: The gemetic seies (Ifiite) with cmm ti me th e becmes divegig d sum is t fixed. 7. Sttemet-: If ithmetic me f tw umbes is /, Gemetic me f the umbes is the hmic me will be 8/. Sttemet-: f gup f umbes (GM) = (AM) (HM). 7. Sttemet-: If, b, c, d be fu distict psitive qutities i H.P. the d > b c, d > bc. Sttemet-: A.M. > G.M. > H.M. 7. Sttemet-: The sum f ithmetic mes betwee tw give umbes is times the sigle ithmetic me betwee them. Sttemet-: th tem f the A.P. with fist tem d cmm diffeece d is ( )d. 7. Sttemet-: If b c = > 0, b > 0, c > 0, the getest vlue f b c = / Sttemet-: If i > 0 i =,,,.., the (... ) f 6
3 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF ANSWER SHEET 9. A 0. B. B. B. D. D. B 6. B 7. C 8. D 9. D 60. D 6. A 6. A 6. A 6. D 6. D 66. D 67. D68. A 69. C 70. C 7. A 7. C 7. A 7. C 7. A IMP QUESTION FROM COMPETETIVE EXAMS. If the gles f qudiltel e i A.P. whse cmm diffeece is 6, 8, 9,0 7, 8, 9,0 0, the the gles f the qudiltel e 6, 7, 8, 9 6, 9,0. If the sum f fist tems f A.P. be equl t the sum f its fist m tems, ( m ), the the sum f its fist ( m ) tems will be [MP PET 98] 0 m m. If p, q, e i A.P. d e psitive, the ts f the qudtic equti px qx = 0 e ll el f [IIT 99] 7 p p 7 < All p d N p d. The sums f tems f thee A.P.'s whse fist tem is d cmm diffeeces e,, e S, S, S espectively. The tue elti is S S = S S S = S S S = S S S = S. The vlue f x stisfyig lg x lg x lg x... lg x = will be x = x = / x = 6. Jim puchsed huse i Rs. 000 d pid Rs. 000 t ce. Rest mey he pmised t py i ul istllmet f Rs. 000 with 0% pe um iteest. Hw much mey is t be pid by Jim [UPSEAT 999] x = Rs. Rs. 07 Rs. 000 Rs Let S, S,... be sques such tht f ech, the legth f side f S equls the legth f digl f S. If the legth f side f S is 0 cm, the f which f the fllwig vlues f is the e f S less the sq cm If S, S, S,..... S m e the sums f tems f m A.P.'s whse fist tems e,,,......, m d cmm diffeeces e,,,....m espectively, the S S S... S = m m ( m ) m ( m ) m ( m ) Ne f the bve 9. If,,,... e i ithmetic pgessi d 0 0 =, the... = [MP PET 999; AMU 997] If the ts f the equti x x 9 x 8 = 0 e i A.P., the thei cmm diffeece will be ± ± ± () ± [UPSEAT 99, 99, 00; RPET 00]. If the fist tem f G.P.,,,.... is uity such tht is lest, the the cmm ti f G.P. is Ne f these. If the sum f the tems f G.P. is S pduct is P d sum f thei ivese is R, th R S S R R S / P is equl t S R, [IIT 966; Rkee 98]. Let (> ) be psitive itege, the the lgest itege m such tht ( ) divides (... ), is [IIT 99] m 7 f 6
4 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF. A G.P. csists f eve umbe f tems. If the sum f ll the tems is times the sum f the tems ccupyig dd plces, the the cmm ti will be equl t. If f (x) is fucti stisfyig f ( x y) = f( x) f( y) f ll x, y N such tht f ( ) = d f( x) = 0. The the vlue x = f is [IIT 99] 6 Ne f these 6. If gemetic mes betwee d b be G,,... G. G... G = G G. G... G = G G G. G... G = G / G. G... G = G / G d gemetic me be G, the the tue elti is 7. α, β e the ts f the equti x x = 0 d γ, δ e the ts f the equti x x b = 0. If α, β, γ, δ fm icesig G.P., the (, b) = [DCE 000] (, ) (, ) (, ) (, 6) = [IIT 98; RPET 99] Ne f these 9. If cs α cs α... =, the α, ( 0 < α < π) is [Rkee 000; AMU 00] π / 8 π / 6 π / π / 0. The fist tem f ifiite gemetic pgessi is x d its sum is. The [IIT Sceeig 00] 0 x 0 0 < x < 0 0 < x < 0 x > 0. If, b, c e i H.P., the the vlue f, is [MP PET 998; Pb. CET 000] b c c b b c Ne f these b b bc c. If m is t f the give equti ( b ) x ( b ) x ( b ) = 0 d m hmic mes e iseted betwee d b, the the diffeece betwee the lst d the fist f the mes equls b b( b ) ( b ) b( b). A by ges t schl fm his hme t speed f x km/hu d cmes bck t speed f y km/hu, the the vege speed is give by [DCE 00] A.M. G.M. H.M. Ne f these. If, b, c, d be i H.P., the c > b d d > b c c bd > b c c bd > b d. If, b, c e the psitive iteges, the ( b)( b c)( c ) is [DCE 000] < 8bc > 8bc = 8bc Ne f these 6. I G.P. the sum f thee umbes is, if is dded t fist tw umbes d subtcted fm thid umbe, the seies becmes A.P., the the getest umbe is [Rkee 97] If, b, c e i G.P. d lg lg b, lg b lg c d lg c lg e i A.P., the, b, c e the legth f the sides f tigle which is Acute gled Obtuse gled Right gled Equiltel 8. If A, A; G, G d H, H be AM ' s, GM ' s d HM ' s betwee tw qutities, the the vlue f GG HH is f 6
5 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF A A H H A A H H A H A H A A H H 9. The hmic me f tw umbes is d the ithmetic d gemetic mes stisfy the elti A G = 7, the umbes e [MNR 987; UPSEAT 999, 000] 6,,,., 0. If the A.M. f tw umbes is gete th G.M. f the umbes by d the ti f the umbes is :, the the umbes e [RPET 988],, 6, Ne f these. If the A.M. d G.M. f ts f qudtic equtis e 8 d espectively, the the qudtic equti will be x 6 x = 0 x 8 x = 0 x 6 x = 0 x 6 x = 0. The A.M., H.M. d G.M. betwee tw umbes e d A.M. espectively e,,,, [Pb. CET 990], d, but t ecessily i this de. The H.M., G.M.,,,, G. If be the ithmetic me f b d c d G, G be the tw gemetic mes betwee them, the G = G G G G G Ne f these G d. Thee umbes fm G.P. If the tem is decesed by 6, the the thee umbes thus btied will cstitute A.P. If the secd tem f this A.P. is decesed by 8, G.P. will be fmed gi, the the umbes will be, 0, 6,, 6, 0, 00 Ne f the bve. If x >, y >, z > e i G.P., the,, I x I y I z e i [IIT 998; UPSEAT 00] A.P. H.P. G.P. Ne f these 6., g, h e ithmetic me, gemetic me d hmic me betwee tw psitive umbes x d y espectively. The idetify the cect sttemet mg the fllwig [Ktk CET 00] h is the hmic me betwee d g N such elti exists betwee, g d h g is the gemetic me betwee d h A is the ithmetic me betwee g d h 7. si θ cs θ is gete th [AMU 000] 8. If, b, c, d e psitive el umbes such tht b c d =, the M = ( b)( c d) stisfies the elti 0 < M M M M 9. Suppse, b, c e i A.P. d, b, c e i G.P. If < b < c d b c =, the the vlue f is [IIT Sceeig 000] [IIT Sceeig 00] f 6
6 Dwld FREE Study Pckge fm & Le Vide Phe : , WhtsApp SEQUENCE & SERIES PART OF 0. th 7 0 tem f the seies... will be. The sum f the seies... t tems is ( ) ( ( ) ) ( ( ) ) Ne f these. F y dd itege, ( ).... ( ) = [IIT 996] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). The sum f tems f the seies... 7 is [UPSEAT 00] ( ). th tem f the seies... will be [Pb. CET 000] 8. The sum f the seies... equls [AMU 00] ( ) ( ) ANSWER b b d 6 c 7 b,c,d 8 9 d 0 c d c c 6 c 7 c 8 c 9 d 0 b c b c c b 6 7 b c c b b c b 6 c 7 d 8 9 d 0 c b d d c d F 9 Yes Que. f IIT-JEE (Advced) & Yes Que. f AIEEE (JEE Mi) we hve ledy distibuted bk 6 f 6
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