Question

Rút gọn tổng: \(S=C\overset{1}{2019}-2^2C\overset{2}{2019}+...-2018^2C\overset{2018}{2019}+2019^2C\overset{2019}{2019}\) bằng:

A. 2019

B. -2019

C. 1

D. 0

Answer 1

Xét khai triển:

\(\left(1-x\right)^{2019}=C_{2019}^0-xC_{2019}^1+x^2C_{2019}^2-...-x^{2019}C_{2019}^{2019}\)

Đạo hàm 2 vế:

\(-2019\left(1-x\right)^{2018}=-C_{2019}^1+2xC_{2019}^2-...-2019x^{2018}C_{2019}^{2019}\)

\(\Rightarrow2019x\left(1-x\right)^{2018}=xC_{2019}^1-2x^2C_{2019}^2+...+2019x^{2019}C_{2019}^{2019}\)

Đạo hàm 2 vế:

\(2019\left(1-x\right)^{2018}-2018.2019x\left(1-x\right)^{2017}=C_{2019}^1-2^2xC_{2019}^2+...+2019^2x^{2018}C_{2019}^{2019}\)

Thay \(x=1\)

\(\Rightarrow0=C_{2019}^1-2^2C_{2019}^2+...+2019^2C_{2019}^{2019}\)

\(\Rightarrow S=0\)