Sequences and Series

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

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

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

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

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

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, 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)

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

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

Let a1, a2, a3, be a G.P. of increasing positive numbers. If a3 a5 = 729 and a2 + a4 = 1114, then 24(a1 + a2 + a3) is equal to

1. 128
2. 129
3. 131
4. 130

Let an be a sequence such that a0 = 0, a1 = 12 and 2an+2 = 5an+1 - 3an, n = 0, 1, 2, 3, . Then k=1100 ak is equal to

1. 3a100 + 100
2. 3a100 - 100
3. 3a99 - 100
4. 3a99 + 100