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

Size: px

Start display at page:

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

Alison Lamb
5 years ago
Views:

1 07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If = a, y = b, z = c, where a, b, c are i A.P. ad = 0 = 0 = 0 l a l <, l b l <, l c l <, the, y, z are i G.P. A.P. Arithmetic-Geometric Progressio H.P. [ AIEEE 005 ] ( ) The sum of the series + e - e ( ) If S = r = 0 C r e + e! + ad t = r = 0 6! e - e C r + 6 6!, the - + d f. is e + e t S - = [ AIEEE 005 ] [ AIEEE 00 ] ( ) Let T r be the rth term of a A.P whose first term is a ad commo differece is d. If for some positive itegers m,, m, T m = ad T = m, the 0 m ( 5 ) The sum of the f rst terms of he series + m is ve. Whe is odd, the sum is ( + ) ( + ) ( + ) ( + ) ( + ) [ AIEEE 00 ] whe is [ AIEEE 00 ] ( 6 ) The sum of the series is!! 6! e - ( e - ) e e - e e - e [ AIEEE 00 ]

2 07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( 7 ) The sum of the series is log e log e log e - log e e [ AIEEE 00 ] ( 8 ) If the sum of the roots of the quadratic equatio a + b + c = 0 is equal to the sum a b c of the squares of their reciprocals, the,, are i c a b A. P. G. P. H. P. A. G. P [ AIEEE 00 ] ( 9 ) The value of terms is ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ( + ) ( + ) + ) ( + ) ( + ) 6 [ AIEEE 00 ] ( 0 ) If the third term of a A. P. s 7 ad its 7th term is more tha three times of its third term, the the sum o its f st 0 terms is ( ) 090 [ AIEEE 00 ] ( ) A ifiite G. P. h s first term ad sum 5, the 0 0 < < 0 < < < 0 { IIT 00 } ( ) If a, a,.. a are positive real umbers whose product is a fied umber c, the the miimum value of a + a + + a - + a is / ( + ) c / c / ( + ) / [ IIT 00 ] ( ) Suppose a, b, c are i A. P. ad a, b, c, are i G. P. If a < b < c ad a + b + c =, the the value of a is - - [ IIT 00 ]

3 07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If the sum of the first terms of the A. P., 5, 8,.., is equal to the sum of the first terms of the A. P. 57, 59, 6,.., the equals 0 [ IIT 00 ] ( 5 ) If the positive umbers a, b, c, d are i A. P., the 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 are positive real umbers such th t a + b + c + d =, the M = ( a + b ) ( c + d ) satisfies the relatio 0 M M M M [ IIT 000 ] ( 7 ) Cosider a ifiite geometric series wi h first erm a ad commo ratio r. If its sum is ad the secod term is, the a d r are, 7 7, 8,, [ IIT 000 ] ( 8 ) Let a, a,, a 0 be A. P. ad h, h,, h 0 be i H. P. If a = h = ad a 0 = h 0 =, the h 7 is 5 6 [ IIT 999 ] ( 9 ) for a positive iteger, a ( ) = , the - a ( 00 ) 00 a ( 00 ) > 00 a ( 00 ) 00 a ( 00 ) > 00 [ IIT 999 ] ( 0 ) Let T r be the rth term of a A. P., for r =,,, If for some positive itegers m,, we have T m = ad T = m, the Tm equals m + 0 [ IIT 998 ] m

4 07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If >, y >, z > are i G. P., the, + l, + l y + l z are i A. P. H. P. G. P. Noe of these [ IT 998 ] ( ) If > is a positive iteger, the the largest iteger m such that ( m + ) divides ( ) is [ IIT 995 ] ( ) The product of positive umbers is uity. The thei sum is a positive iteger divisible by equal to + ever less tha [ IIT 99 ] 7 5 ( ) The sum of terms of the se es is equal to [ IIT 988 ] ( 5 ) If the first ad he ( - )th terms of a A. P., G. P. ad H. P. are equal ad their th terms are a,, c respectively, the a = b = c a b c a + c = b ac - b = 0 [ IIT 988 ] ( 6 ) If a,, c, d ad p are distict real umbers such that ( + b + c ) p - ( ab + bc + cd ) p + ( b + c + d ) 0, the a, b, c ad d are i A. P. are i G. P. are i H. P. satisfy ab = cd [ IIT 987 ] ( 7 ) If a, b, c are i G. P., the the equatios a + b + c = 0 ad d + e + f = 0 d e f have a commo root if,, are i a b c AP GP HP oe of these [ IIT 985 ]

5 07 - SEQUENCES AND SERIES Page 5 ( Aswers at he ed of all questios ) ( 8 ) The third term of a geometric progressio is. The product of the first five terms is 5 oe of these [ IIT 98 ] ( 9 ) If,,.., are ay real umbers ad is ay positive iteger, t e i < i i = i = i i i = i = i i = i i = oe of these [ IIT 98 ] ( 0 ) If, y ad z are the p th, q th ad r th erm respectively of a A. P. ad also of a G. P., the y - z y z - z - y is equal to ( ) yz 0 oe of these [ IIT 979 ] +, - ad - are cosecutive terms of a series i H. P. G. P. A. P. A. P., G. P. ( ) If S = P + ( - ) Q, where S deotes the sum of the first terms of a A. P the the commo differece is a ) P + Q P + Q Q Q ( ) If S = + + +, where S deotes the sum of the first terms of a series ad t m = 9, the m = 0 ( ) If the first term mius third term of a G. P. = 768 ad the third term mius seveth term of the same G. P. = 0, the the product of first terms =

6 07 - SEQUENCES AND SERIES Page 6 ( Aswers at he ed of all questios ) ( 5 ) If the sequece a, a, a, a form a A. P., the a - a + a - + a - a = ( a a - ) - ( a a + ) + ( 6 ) If T r deotes rth term of a H. P. ad ( a a - ) - Noe of these T - T T6 - T9 = 7, t e T - T5 T - T8 ( 7 ) The sum of ay te positive real umbers multiplied by the sum of their reciprocals is ( 8 ) If S deotes the sum of ir t terms of a A. P. ad S = S, the the ratio S is equal to S ( 9 ) If a, b, c a e three uequal positive quatities i H. P., the a 0 c 0 < b 0 a 0 + c 0 < b 0 + c < b oe of these Aswers 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 =

Objective Mathematics

Objective Mathematics . If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic