Evaluating $\sum\limits_{i=0}^n{n\choose i}2^i$

Question

I'm being asked to "evaluate the sum"

$$\sum\limits_{i=0}^n{n\choose i}2^i$$

Writing out the first few values of $n$ it is clear that the sum is equal to $3^n$, however I'm not sure how to prove this. I know that it is straight forward to show that $\sum\limits_{i=0}^n{n\choose i}=2^n$, we just set $x=y=1$ in the binomial formula, but the $2^i$ seems to make things much less obvious.

The wording of the question seems ambiguous. I'm not sure if "evaluate" means that I don't need to prove the result? But just evaluating a few values of $n$ and calling it a day seems insufficient.

If anybody could give me a hint to maybe set me on the right track that would be great. Thanks in advance!

Answer 1

HINT

Rewrite the given expression as \begin{align*} \sum_{i=0}^{n}{n\choose i}2^{i} = \sum_{i=0}^{n}{n\choose i}2^{i}1^{n-i} \end{align*}

and apply the binomial theorem.

Can you take it from here?