Question 81
$$
2 \sum_{k=0}^{n} \binom{2n}{k} (n-k) = \binom{2n}{n+1} (n+1), \ n = 1, 2, \dots
$$
and find the value of
$$
\sum_{k=0}^{n} k \binom{2n}{k}.
$$
2 \sum_{k=0}^{n} \binom{2n}{k} (n-k) = \binom{2n}{n+1} (n+1), \ n = 1, 2, \dots
$$
and find the value of
$$
\sum_{k=0}^{n} k \binom{2n}{k}.
$$