How do I derive a formula for summation of this series?

Question

I have a series in which the general term is represented by

$a_n = \frac{k^{n+1} - k - nk +n}{(k-1)^2} $

where $k$ is a constant.

How do I derive the formula for $\sum_{n=1}^{n=b} S_n$ (summation of n terms)?

I'm not really good at mathematics (high school level), so kindly explain it in the simplest way possible.

Answer 1

HINT

Note that you can divide and sum each part separately

$$a_n = \frac{k^{n+1} - k - nk +n}{(k-1)^2}=\frac{k^{n+1} - k }{(k-1)^2}-\frac{nk -n}{(k-1)^2}= \frac{k^{n+1} }{(k-1)^2}-\frac{k }{(k-1)^2}-\frac{n}{(k-1)}$$

and

$$\sum_{n=1}^{n=b} k^{n+1}=\sum_{n=0}^{n=b-1} k^{n+2}=k^2\sum_{n=0}^{n=b-1} k^{n}=k^2\frac{1-k^b}{1-k}$$

$$\sum_{n=1}^{n=b} n=\frac{b(b+1)}{2}$$