Let S_n denote the sum of the first n terms of an arithmetic progression whose first term is 3 and common difference is d . If _ n=1 ^ S_n 4^n = 2 , then the value of d is :

Question

Quantrex Academy · Accepted Answer

Correct answer: C. 9 2

The sum of the first n terms of the AP is given by: S_n = n 2 [2(3) + (n-1)d] = 3n + d 2 n(n-1) We are given that _ n=1 ^ S_n 4^n = 2 . Substituting S_n into the series: _ n=1 ^ ( 3n 4^n + d 2 n(n-1) 4^n ) = 2 This can be split into two standard infinite summations. Let x = 1 4 . The first sum is 3 _ n=1 ^ n x^n . We know that _ n=1 ^ n x^n = x (1-x)^2 . For x = 1 4 , this evaluates to 1/4 (3/4)^2 = 4 9 . So, 3 _ n=1 ^ n 4^n = 3 ( 4 9 ) = 4 3 . The second sum is d 2 _ n=1 ^ n(n-1) x^n . We know that _ n=1 ^ n(n-1)

Let S_n denote the sum of the first n terms of an arithmetic progression whose first term is 3 and common difference is d . If _ n=1 ^ S_n 4^n = 2 , then the value of d is :

Options

Correct answer

Step-by-step solution

More from Sequences and Series