Sequences and Series

If 7=5+ 17(5+ )+ 172(5+2 )+ 173(5+3 )+ , then the value of is :

1. 17
2. 1
3. 67
4. 6

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

1. -120
2. -1200
3. -1080
4. -1020

The roots of the quadratic equation 3x2-px+q=0 are 10th and 11th terms of an arithmetic progression with common difference 32. If the sum of the first 11 terms of this arithmetic progression is 88, then q-2p is equal to .

Let a1,a2,a3, be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to:

1. 812
2. 784
3. 628
4. 526

If r=1nTr= (2n-1)(2n+1)(2n+3)(2n+5)64, then n r=1n ( 1Tr ) is equal to :

1. 23
2. 13
3. 1
4. 0

Suppose that the number of terms in an A.P. is 2k, k . If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to:

1. 8
2. 6
3. 4
4. 5

If the sum of the series 11 (1 + d) + 1(1 + d)(1 + 2d) + + 1(1 + 9d)(1 + 10d) is equal to 5, then 50d is equal to :

1. 5
2. 10
3. 15
4. 20

Let a, ar, ar2, be an infinite G.P. If n=0 arn = 57 and n=0 a3r3n = 9747, then a + 18r is equal to

1. 27
2. 38
3. 31
4. 46

If ( 1 + 1 + 1 + 2 + + 1 + 1012 ) - ( 12 1 + 14 3 + 16 5 + + 12024 2023 ) = 12024, then is equal to .

Let the positive integers be written in the form : [figure] If the kth row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is .