Question

Tìm n thuộc Z biết 

\(2.2^2+3.2^3+4.2^4+5.2^5+...+n.2^n=2^{n+10}\)

Answer 1

Đặt \(A=2.2^2+3.2^3+4.2^4+5.2^5+...+n.2^n\)

\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)

\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)

\(-2.2^2-3.2^3-4.2^4-5.2^5-...-n.2^n\)

\(A=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^n\right)\)

Đặt \(M=\left(2^3+2^4+...+2^n\right)\)

\(\Rightarrow2M=\left(2^4+2^5+...+2^{n+1}\right)\)

\(\Rightarrow M=2^{n+1}-2^3\)

\(\Rightarrow A=n.2^{n+1}-2^3-2^{n+1}+2^3\)

\(\Rightarrow A=\left(n-1\right)2^{n+1}=2^{n+10}\)

\(\Rightarrow\left(n-1\right)=2^9\)

\(\Rightarrow n=513\)