summation - Sum to the nth term of an arithmetic-geometric series

We have to find the sum to nth term in the following series:

$$1-\frac{2}{2}+\frac{3}{2^2}-\frac{4}{2^3}+$$ up to n^th term.

I tried using the common method of successive differences. It lead me to an answer that was:

$$\frac{3}{2}S=\frac{(-2)^n2+4}{3}+\frac{2-3n}{(-2)^n}$$

Where S is the sum of the series.

I'm not sure if this is the answer. Could anyone help me out by checking this answer and recommend a better, not so sophisticated method.