Simplifying This Summation - Need Help

Question

For each $m \in \mathbb{Z}^+$, let $S_m$ = $\sum_{k=0}^m 2^k{m \choose k}$.

If $m \in \mathbb{Z}^+$, determine the value of $$\frac{S_{n+1}}{S_n}$$

So far, I have simplified it down to this: https://i.stack.imgur.com/Spf6t.jpg

I don't know how to go from here.

Answer 1

Note that according to the binomial theorem you have $$\sum_{k=0}^m 2^k{m \choose k} = (2+1)^m$$