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

Using the hint, the summand can be transformed as 
$$(k+1){n\choose k}=k{n\choose k}+{n\choose k}=n{n-1\choose k-1}+{n\choose k}.$$
Thus we obtain
$$\sum_{k=0}^n(k+1){n\choose k} = n\sum {n-1\choose k-1}+\sum {n\choose k} = n2^{n-1}+2^n. $$
