Let S_n be the sum of the first n terms of an arithmetic progression whose first term is 2 and common difference is 3 . If T = _ n=1 ^ S_n 3^n , then the value of 8T is

Step-by-step solution

The first term of the arithmetic progression is a = 2 and the common difference is d = 3 . The sum of the first n terms is given by S_n = n 2 [2a + (n - 1)d] = n 2 [4 + (n - 1)3] = 3n^2 + n 2 We need to evaluate T = _ n=1 ^ S_n 3^n = 1 2 _ n=1 ^ 3n^2 + n 3^n Let T₁ = _ n=1 ^ n 3^n = 1 3 + 2 3^2 + 3 3^3 + 1 3 T₁ = 1 3^2 + 2 3^3 + Subtracting the two equations: 2 3 T₁ = 1 3 + 1 3^2 + 1 3^3 + = 1 3 1 - 1 3 = 1 2 T₁ = 3 4 Let T₂ = _ n=1 ^ n^2 3^n = 1 3 + 4 3^2 + 9 3^3 + 1 3 T₂ = 1 3^2 + 4 3^3 + Subtracting the two equa