Question

Rút gọn tổng: \(S=C\overset{1}{n}+1.2C\overset{2}{n}+2.3C\overset{3}{n}+...+\left(n-1\right)nC\overset{n}{n}\) bằng:

A. \(\left(n-1\right)n.2^{n-2}\)

B. \(n.2^{n-2}\)

C. \(\left(n-1\right)n.2^{n-1}+n\)

D. \(\left(n-1\right)n.2^{n-2}+n\)

Answer 1

Xét khai triển:

\(\left(1+x\right)^n=C_n^0+xC_n^1+x^2C_n^2+...+x^nC_n^n\)

Đạo hàm 2 vế:

\(n\left(1+x\right)^{n-1}=C_n^1+2xC_n^2+...+nx^{n-1}C_n^n\)

Tiếp tục đạo hàm 2 vế:

\(\left(n-1\right)n\left(1+x\right)^{n-2}=2C_n^2+2.3xC_n^3+...+\left(n-1\right)nx^{n-2}C_n^n\)

Thay \(x=1\)

\(\Rightarrow\left(n-1\right)n.2^{n-2}=1.2C_n^2+2.3C_n^3+...+\left(n-1\right)nC_n^n\)

\(\Rightarrow\left(n-1\right)n.2^{n-2}+n=C_n^1+1.2C_n^2+...+\left(n-1\right)n.C_n^n\)

\(\Rightarrow S=\left(n-1\right)n.2^{n-2}+n\)