Sequences and Series

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

If three successive terms of a G.P. with common ratio r(r>1) are the lengths of the sides of a triangle and [r] denotes the greatest integer less than or equal to r, then 3[r]+[-r] is equal to .

The sum of the series 11-3 12+14+ 21-3 22+24+ 31-3 32+34+ up to 10-terms is

1. 45109
2. - 55109
3. 55109
4. - 45109

Let 2nd, 8th and 44th terms of a non-constant A. P. be respectively the 1st, 2nd and 3rd terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

1. 990
2. 980
3. 960
4. 970

Let Sn denote the sum of first n terms of an arithmetic progression. If S20=790 and S10=145, then S15-S5 is :

1. 405
2. 390
3. 410
4. 395