: Tính các tổng sau:${{S}_{2}}=C_{n}^{1}+2C_{n}^{2}+…+nC_{n}^{n}$

: Tính các tổng sau:${{S}_{2}}=C_{n}^{1}+2C_{n}^{2}+…+nC_{n}^{n}$

C. $2n{{.2}^{n-1}}$

B. $n{{.2}^{n+1}}$

C. $2n{{.2}^{n+1}}$

D. $n{{.2}^{n-1}}$

Hướng dẫn

Chọn D

Ta có: $kC_{n}^{k}=k.\frac{n!}{k!(n-k)!}=\frac{n!}{(k-1)!\text{ }\!\![\!\!\text{ }(n-1)-(k-1)\text{ }\!\!]\!\!\text{ }!}$

$=n\frac{(n-1)!}{(k-1)!\text{ }\!\![\!\!\text{ }(n-1)-(k-1)\text{ }\!\!]\!\!\text{ }!}=nC_{n-1}^{k-1}$, $\forall k\ge 1$

$\Rightarrow {{S}_{2}}=\sum\limits_{k=1}^{n}{nC_{n-1}^{k-1}}=n\sum\limits_{k=0}^{n-1}{C_{n-1}^{k}}=n{{.2}^{n-1}}$.