Question

tính tổng 

Answer 1

Xét khai triển:

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

\(\Rightarrow x\left(1+x\right)^{2019}=xC_{2019}^0+x^2C_{2019}^1+x^3C_{2019}^2+...+x^{2020}C_{2019}^{2019}\)

Đạo hàm 2 vế:

\(\Rightarrow\left(1+x\right)^{1019}+2019x\left(1+x\right)^{2018}=C_{2019}^0+2xC_{2019}^1+3x^2C_{2019}^2+...+2020x^{2019}C_{2019}^{2019}\)

Thay \(x=1\)

\(\Rightarrow2^{2019}+2019.2^{2018}=C_{2019}^0+2C_{2019}^1+...+2020C_{2019}^{2019}\)

\(\Rightarrow2021.2^{2019}=C_{2019}^0+2C_{2019}^1+...+2020C_{2019}^{2019}\)