(a) Prove that, for $n \ge 1$,
$\binom{n}{0} + \frac{1}{2} \binom{n}{1} + \frac{1}{3} \binom{n}{2} + \dots + \frac{1}{i+1} \binom{n}{i} + \dots + \frac{1}{n+1} \binom{n}{n} = \frac{2^{n+1} - 1}{n+1}$.
(b) Prove that, for $n \ge 1$,
$\frac{1}{2} \binom{n}{0} + \frac{1}{3} \binom{n}{1} + \frac{1}{4} \binom{n}{2} + \dots + \frac{1}{i+2} \binom{n}{i} + \dots + \frac{1}{n+2} \binom{n}{n} = \frac{n2^{n+1} + 1}{(n+1)(n+2)}$.