Simplify $\sum_{i=0}^n (i+1)\binom ni$ Simplifying this expression$$1\cdot\binom{n}{0}+ 2\cdot\binom{n}{1}+3\cdot\binom{n}{2}+ \cdots+(n+1)\cdot\binom{n}{n}= ?$$
$$\text{Hint: } \binom{n}{k}= \frac{n}{k}\cdot\binom{n-1}{k-1} $$

\begin{align}\sum_{k=0}^n(k+1) \binom{n}{k}&=\sum_{k=0}^nk \binom{n}{k}+\sum_{k=0}^n \binom{n}{k}=n\sum_{k=1}^n \binom{n-1}{k-1}+2^n\\&=n\sum_{k=0}^{n-1} \binom{n-1}{k}+2^n=n2^{n-1}+2^n
\end{align}
