Giúp dùm nhỏ toán 11

[nhonho1234]

\[a=c^{1}_{2011}+c^3_{2011}+c^5_{2011}+c^7_{2011} +...+c^{2011}_{2011}\]

 

[NguoiDien]

\[A=C^{1}_{2011}+C^3_{2011}+C^5_{2011}+C^7_{2011} +...+C^{2011}_{2011}\]

Dùng khai triển nhị thức Niu-Ton:

\[(1+1)^{2011}=C_{2011}^{0}.1^{0}.1^{2011}+C_{2011}^{1}.1^{1}.1^{2010}+C_{2011}^{2}.1^{2}.1^{2009}+...+C_{2011}^{2010}.1^{2010}.1^{1}+C_{2011}^{0}.1^{2011}.1^{0}\]

Suy ra:

\[2^{2011}=2(C^{1}_{2011}+C^3_{2011}+C^5_{2011}+C^7_{2011} +...+C^{2011}_{2011})\]

\[\Rightarrow C^{1}_{2011}+C^3_{2011}+C^5_{2011}+C^7_{2011} +...+C^{2011}_{2011}=\frac{2^{2011}}{2}=2^{2010}\]