Binomial Expansion Find: $\sum_{k=0}^n\binom{n}{k} 2^k $

Question

Find: $\sum_{k=0}^n \binom{n}{k}2^k$

So I know how to find the coefficients of binomial expansions but I’m not sure how to do this type of problem where I just find the summation.

The example in the book is $\sum_{k=0}^n \binom{n}{k} 4^k $ and the answer is $5^n$

Thank you for any and all help.

Answer 1

$$\sum_{k=0}^n \binom{n}{k}2^k =\sum_{k=0}^n \binom{n}{k}2^k 1^{n-k}=(2+1)^n$$ By Binomial theorem.

$$(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^kb^{n-k}$$

Answer 2

Hint:

rewrite is as $$\sum_{k=0}^n\binom{n}{k}2^k1^{n-k}$$ Any recognition?