Question

Tính tổng sau: \(C^1_{2021}+C^3_{2021}+C^5_{2021}+...+C^{2017}_{2021}+C^{2019}_{2021}.\)

Answer 1

Xét khai triển:

\(2^{2021}=\left(1+1\right)^{2021}=C_{2021}^0+C_{2021}^1+...+C_{2021}^{2020}+C_{2021}^{2021}\) (1)

\(0=\left(1-1\right)^{2021}=C_{2021}^0-C_{2021}^1+C_{2021}^2+...+C_{2021}^{2020}-C_{2021}^{2021}\) (2)

Trừ vế cho vế (1) và (2):

\(2^{2021}=2.C_{2021}^1+2.C_{2021}^3+...+2C_{2021}^{2021}\)

\(\Rightarrow C_{2021}^1+C_{2021}^3+...+C_{2021}^{2019}+C_{2021}^{2021}=\dfrac{2^{2021}}{2}=2^{2020}\)

\(\Rightarrow C_{2021}^1+C_{2021}^3+...+C_{2021}^{2019}+1=2^{2020}\)

\(\Rightarrow C_{2021}^1+C_{2021}^3+...+C_{2021}^{2019}=2^{2020}-1\)