07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n

Transcription

1 07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l <, l b l <, l c l <, th, y, z ar i G.P. A.P. Arithmtic-Gomtric Progrssio H.P. [ AIEEE 005 ] ( ) Th sum of th sris + - ( ) If S = r = 0 C r +! + 6! - ad t = r = 0 C r + 6 6!, th - + d f. is + t S - = [ AIEEE 005 ] [ AIEEE 00 ] ( ) Lt T r b th rth trm of a A.P whos first trm is a ad commo diffrc is d. If for som positiv itgrs m,, m, T m = ad T = m, th 0 m ( 5 ) Th sum of th f rst trms of h sris + m is v. Wh is odd, th sum is ( + ) ( + ) ( + ) ( + ) ( + ) [ AIEEE 00 ] wh is [ AIEEE 00 ] ( 6 ) Th sum of th sris is!! 6! - ( - ) - - [ AIEEE 00 ]

2 07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( 7 ) Th sum of th sris is log log log - log [ AIEEE 00 ] ( 8 ) If th sum of th roots of th quadratic quatio a + b + c = 0 is qual to th sum a b c of th squars of thir rciprocals, th,, ar i c a b A. P. G. P. H. P. A. G. P [ AIEEE 00 ] ( 9 ) Th valu of trms is ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ( + ) ( + ) + ) ( + ) ( + ) 6 [ AIEEE 00 ] ( 0 ) If th third trm of a A. P. s 7 ad its 7th trm is mor tha thr tims of its third trm, th th sum o its f st 0 trms is ( ) 090 [ AIEEE 00 ] ( ) A ifiit G. P. h s first trm ad sum 5, th 0 0 < < 0 < < < 0 { IIT 00 } ( ) If a, a,.. a ar positiv ral umbrs whos product is a fid umbr c, th th miimum valu of a + a + + a - + a is / ( + ) c / c / ( + ) / [ IIT 00 ] ( ) Suppos a, b, c ar i A. P. ad a, b, c, ar i G. P. If a < b < c ad a + b + c =, th th valu of a is - - [ IIT 00 ]

3 07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If th sum of th first trms of th A. P., 5, 8,.., is qual to th sum of th first trms of th A. P. 57, 59, 6,.., th quals 0 [ IIT 00 ] ( 5 ) If th positiv umbrs a, b, c, d ar i A. P., th abc, abd, acd, bcd ar ot i A. P. / G. P. / H. P. i A. P. i G. P. i H P. [ IIT 00 ] ( 6 ) If a, b, c, d ar positiv ral umbrs such th t a + b + c + d =, th M = ( a + b ) ( c + d ) satisfis th rlatio 0 M M M M [ IIT 000 ] ( 7 ) Cosidr a ifiit gomtric sris wi h first rm a ad commo ratio r. If its sum is ad th scod trm is, th a d r ar, 7 7, 8,, [ IIT 000 ] ( 8 ) Lt a, a,, a 0 b A. P. ad h, h,, h 0 b i H. P. If a = h = ad a 0 = h 0 =, th h 7 is 5 6 [ IIT 999 ] ( 9 ) for a positiv itgr, a ( ) = , th - a ( 00 ) 00 a ( 00 ) > 00 a ( 00 ) 00 a ( 00 ) > 00 [ IIT 999 ] ( 0 ) Lt T r b th rth trm of a A. P., for r =,,, If for som positiv itgrs m,, w hav T m = ad T = m, th Tm quals m + 0 [ IIT 998 ] m

4 07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If >, y >, z > ar i G. P., th, + l, + l y + l z ar i A. P. H. P. G. P. No of ths [ IT 998 ] ( ) If > is a positiv itgr, th th largst itgr m such that ( m + ) divids ( ) is [ IIT 995 ] ( ) Th product of positiv umbrs is uity. Th thi sum is a positiv itgr divisibl by qual to + vr lss tha [ IIT 99 ] 7 5 ( ) Th sum of trms of th s s is qual to [ IIT 988 ] ( 5 ) If th first ad h ( - )th trms of a A. P., G. P. ad H. P. ar qual ad thir th trms ar a,, c rspctivly, th a = b = c a b c a + c = b ac - b = 0 [ IIT 988 ] ( 6 ) If a,, c, d ad p ar distict ral umbrs such that ( + b + c ) p - ( ab + bc + cd ) p + ( b + c + d ) 0, th a, b, c ad d ar i A. P. ar i G. P. ar i H. P. satisfy ab = cd [ IIT 987 ] ( 7 ) If a, b, c ar i G. P., th th quatios a + b + c = 0 ad d + + f = 0 d f hav a commo root if,, ar i a b c AP GP HP o of ths [ IIT 985 ]

5 07 - SEQUENCES AND SERIES Pag 5 ( Aswrs at h d of all qustios ) ( 8 ) Th third trm of a gomtric progrssio is. Th product of th first fiv trms is 5 o of ths [ IIT 98 ] ( 9 ) If,,.., ar ay ral umbrs ad is ay positiv itgr, t i < i i = i = i i i = i = i i = i i = o of ths [ IIT 98 ] ( 0 ) If, y ad z ar th p th, q th ad r th rm rspctivly of a A. P. ad also of a G. P., th y - z y z - z - y is qual to ( ) yz 0 o of ths [ IIT 979 ] +, - ad - ar coscutiv trms of a sris i H. P. G. P. A. P. A. P., G. P. ( ) If S = P + ( - ) Q, whr S dots th sum of th first trms of a A. P th th commo diffrc is a ) P + Q P + Q Q Q ( ) If S = + + +, whr S dots th sum of th first trms of a sris ad t m = 9, th m = 0 ( ) If th first trm mius third trm of a G. P. = 768 ad th third trm mius svth trm of th sam G. P. = 0, th th product of first trms =

6 07 - SEQUENCES AND SERIES Pag 6 ( Aswrs at h d of all qustios ) ( 5 ) If th squc a, a, a, a form a A. P., th a - a + a - + a - a = ( a a - ) - ( a a + ) + ( 6 ) If T r dots rth trm of a H. P. ad ( a a - ) - No of ths T - T T6 - T9 = 7, t T - T5 T - T8 ( 7 ) Th sum of ay t positiv ral umbrs multiplid by th sum of thir rciprocals is ( 8 ) If S dots th sum of ir t trms of a A. P. ad S = S, th th ratio S is qual to S ( 9 ) If a, b, c a thr uqual positiv quatitis i H. P., th a 0 c 0 < b 0 a 0 + c 0 < b 0 + c < b o of ths Aswrs d d a a b b d c c c b a d c d a d d a,d c b c d c b,d b a b d c c d a a a b c b d =