How to derive the simplest forms of $\sum{\binom{n}{i} \times 2^{n-i}}$ and $\sum{i \times \binom{n}{i} \times 2^{n-i}}$?
Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$