Sequences and Series

Let a1, a2, a3, be in an arithmetic progression of positive terms. Let Ak = a12 - a22 + a32 - a42 + + a2k-12 - a2k2. If A3 = -153, A5 = -435 and a12 + a22 + a32 = 66, then a17 - A7 is equal to .

For x 0, the least value of K, for which 41 + x + 41 - x, K2, 16x + 16-x are three consecutive terms of an A.P., is equal to :

1. 10
2. 4
3. 8
4. 16

If 1 + 3 - 22 3 + 5 - 2 618 + 9 3 - 11 236 3 + 49 - 20 6180 + upto = 2 + ( ba + 1 ) e ( ab ), where a and b are integers with (a, b) = 1, then 11a + 18b is equal to .

Let the first three terms 2, p and q, with q 2, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to:

1. 151
2. 177
3. 163
4. 169

Let a=1+ 2C23!+ 3C24!+ 4C25!+ , b=1+ 1C0+1C11!+ 2C0+2C1+2C22! + 3C0+3C1+3C2+3C33!+ . Then 2ba2 is equal to .

The value of 1 22 + 2 32 + + 100 (101)212 2 + 22 3 + + 1002 101 is

1. 305301
2. 306305
3. 3231
4. 3130

Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the arithmetic mean of a, b and c is 8, then the cube of the geometric mean of a, b and c is

1. 120
2. 316
3. 312
4. 128

Let 3, a, b, c be in A.P. and 3, a-1, b+1, c+9 be in G.P. Then, the arithmetic mean of a, b and c is :

1. -4
2. -1
3. 13
4. 11

Let 3, 7, 11, 15, , 403 and 2, 5, 8, 11, , 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to .

Let Sn denote the sum of the first n terms of an arithmetic progression. If S10=390 and the ratio of the tenth and the fifth terms is 15:7, then S15-S5 is equal to :

1. 800
2. 890
3. 790
4. 690