show that $\sum\limits_{k=0}^n \binom{n}{k}2^k=\sum\limits_{k=0}^n \binom{n}{k}2^k\cdot 1^{n-k}= 3^n$

Question

Show that $$\sum\limits_{k=0}^n \binom{n}{k}2^k=\sum\limits_{k=0}^n \binom{n}{k}2^k\cdot 1^{n-k}= 3^n$$

I know this true but i really having a hard time arrive there. Is it just that $(2+1)^n = 3^n$?

Answer 1

Yes indeed, you are just writing the binomial expansion of $(2+1)^n$.