Question

Find the sum to n terms of the series 2+3.3+4.3^2+5.3^3+.....?

Answer 1

See below.

Explanation:

`sum_(k=0)^n(k+2)x^k = xsum_(k=0)^nkx^(k-1)+2sum_(k=0)^n x^k`

but

`sum_(k=0)^n x^k = (x^(n+1)-1)/(x-1)`

and

`sum_(k=0)^n k x^(k-1) = d/(dx) sum_(k=0)^n x^k = (1 + (n (x-1)-1) x^n)/(x-1)^2` then

`sum_(k=0)^n(k+2)x^k = x( (1 + (n (x-1)-1) x^n)/(x-1)^2)+ 2( (x^(n+1)-1)/(x-1))`

now making `x = 3` we have

`sum_(k=0)^n(k+2)3^k=1/4 (3^(n+1) (3 + 2 n)-1)`