Sequences and Series

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

The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + + (x + 2n - 2)(x + 2n) = 8n3 are two consecutive even integers, is :

1. 3
2. 6
3. 9
4. 12

Let a1, a22, a322, , a1029 be a G.P. of common ratio 1 2. If a1 + a2 + + a10 = 62, then a1 is equal to:

1. 2 - 1
2. 2( 2 - 1)
3. 2 - 2
4. 2(2 - 2)

If 114 + 124 + 134 + = 490, 114 + 134 + 154 + = , 124 + 144 + 164 + = , then is equal to :

1. 23
2. 14
3. 18
4. 15

Let x1, x2, x3, x4 be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x1, x2, x3, x4, then the resulting numbers are in an arithmetic progression. Then the value of 124(x1 x2 x3 x4) is:

1. 18
2. 216
3. 36
4. 72

Let an be the nth term of an A.P. If Sn = a1 + a2 + a3 + + an = 700, a6 = 7 and S7 = 7, then an is equal to :

1. 65
2. 56
3. 70
4. 64

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :

1. 757
2. 755
3. 750
4. 760

1 + 3 + 52 + 7 + 92 + up to 40 terms is equal to

1. 40870
2. 41880
3. 43890
4. 33980

If the sum of the first 20 terms of the series 4 14 + 3 12 + 14 + 4 24 + 3 22 + 24 + 4 34 + 3 32 + 34 + 4 44 + 3 42 + 44 + is mn, where m and n are coprime, then m + n is equal to :

1. 423
2. 421
3. 422
4. 420

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of A.P.'s in A and B respectively such that D=d+3, d>0. If p+qp-q= 195, then p-q is equal to

1. 540
2. 450
3. 600
4. 630