Sum of powers of 2

#power sum#2^n#binomial

Statement

Compute $\sum_{k=0}^{n}2^k\binom{n}{k}$ .

Hint

Set $a=1$ , $b=2$ in the binomial theorem.

Show solutionHide solution

Binomial theorem. $(1+2)^n=\sum_k\binom{n}{k}2^k$ .

Conclusion. $\sum_k 2^k\binom{n}{k}=3^n$ .