Sequences and Series

Let =3+4+8+9+13+14+ upto 40 terms. If ( ) 1020 is a root of the equation x2+x-2=0, (0, 2 ), then 2 +3 2 is equal to :

1. 2
2. 74
3. 52
4. 32

The value of 13-23+33- +153 is :

1. 1706
2. 1856
3. 1982
4. 2403

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is :

1. 349
2. 3413
3. 329
4. 3213

For the functions f( )= 2 + 2 , and g( )= 2 + 2 , > >0, let 0< < 2 f( )= 0< < g( ). If the first term of a G.P. is ( 2 ), its common ratio is ( 2 ) and the sum of its first 10 terms is mn, (m, n)=1, then m+n is equal to .

Consider the quadratic equation (n2-2n+2 )x2-3x+ (n2-2n+2 )2=0, n R. Let be the minimum value of the product of its roots and be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is and the common ratio is , is :

1. 6137
2. 12181
3. 364243
4. 1093729

The sum 1+ 12 (12+22 )+ 13 (12+22+32 )+ upto 10 terms is equal to :

1. 130
2. 155
3. 3152
4. 3252

Let the sum of the first n terms of an A.P. be 3n2+5n. Then the sum of squares of the first 10 terms of the A.P. is:

1. 10220
2. 12860
3. 15220
4. 19780

n=110 ( 528n(n+1)(n+2) ) is equal to:

1. 65
2. 130
3. 220
4. 440

If the sum of the first 10 terms of the series 11+14 4+ 21+24 4 + 31+34 4+ 41+44 4+ is mn, (m, n)=1, then m+n is equal to :

1. 256
2. 264
3. 276
4. 284

Let A1, A2, A3, , A39 be 39 arithmetic means between the numbers 59 and 159. Then the mean of A25, A28, A31 and A36 is equal to :

1. 129
2. 136
3. 131.50
4. 134

The first term of an A.P. of 30 non-negative terms is 103. If the sum of this A.P. is the cube of its last term, then its common difference is:

1. 587
2. 2583
3. 1529
4. 529

Let = 14+ 18+ 116+ and = 13+ 19+ 127+ . Then the value of (0.2) 5( )+(0.04) 5( ) is equal to :

1. 4
2. 5
3. 8
4. 25

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A B, which are divisible by 3, is :

1. 4
2. 5
3. 6
4. 7

If k=1n ak=6n3, then k=16 ( ak+1-ak36 )2 is equal to .

Let a1, a2, a3, be an A.P. and g1=a1, g2, g3, be an increasing G.P. If a1=a2+g2=1 and a3+g3=4, then a10+g5 is equal to:

1. 81
2. 76
3. 62
4. 55

The sum 131+ 13+231+3+ 13+23+331+3+5+ up to 8 terms, is :

1. 70
2. 71
3. 72
4. 73

The value of k=1 (-1)k+1 ( k(k+1)k! ) is

1. e/2
2. e
3. 2/e
4. 1/e

The common difference of the A.P.: a1, a2, , am is 13 more than the common difference of the A.P.: b1, b2, , bn. If b31 = -277, b43 = -385 and a78 = 327, then a1 is equal to

1. 21
2. 19
3. 24
4. 16

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is R - (a, b), then a2 + b2 is equal to .

6326 + 10 1325 + 10 2324 + 10 22323 + + 10 2243 is equal to :

1. 226
2. 325
3. 326
4. 225

If r=125 ( rr4 + r2 + 1 ) = pq, where p and q are positive integers such that (p, q) = 1, then p + q is equal to .

Let 729, 81, 9, 1, be a sequence and Pn denote the product of the first n terms of this sequence. If 2 n=140(Pn) 1n = 3 -13 and ( , ) = 1, then + is equal to

1. 73
2. 74
3. 75
4. 76

Consider an A.P.: a1, a2, , an; a1 > 0. If a2 - a1 = -34, an = 14a1, and i=1n ai = 5252, then i=117 ai is equal to

1. 238
2. 136
3. 476
4. 952

Let a1, a2, a3, a4 be an A.P. of four terms such that each term of the A.P. and its common difference l are integers. If a1 + a2 + a3 + a4 = 48 and a1a2a3a4 + l4 = 361, then the largest term of the A.P. is equal to

1. 27
2. 24
3. 23
4. 21

( 13 + 47 ) + ( 132 + 13 47 + 4272 ) + ( 133 + 132 47 + 13 4272 + 4373 ) + up to infinite terms, is equal to

1. 74
2. 43
3. 65
4. 52

Let k=1n ak = n2 + n. If a10 = 59 and a6 = 7a1, then + is equal to :

1. 3
2. 5
3. 7
4. 12

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

1. -26
2. -20
3. -24
4. -22

Suppose a, b, c are in A.P. and a2, 2b2, c2 are in G.P. If a < b < c and a + b + c = 1, then 9(a2 + b2 + c2) is equal to .

Let a1, a2, a3, be a G.P. of increasing positive terms such that a2 a3 a4 = 64 and a1 + a3 + a5 = 8137. Then a3 + a5 + a7 is equal to :

1. 3256
2. 3252
3. 3248
4. 3244

Let a1 = 1 and for n 1, an+1 = 12an + n2 - 2n - 1n2(n + 1)2. Then | n=1 (an - 2n2 ) | is equal to .