Tìm số tự nhiên n thoả mãn 2.2^2+3.2^3+4.2^4+...+n.2^n=2^n+11

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

Ta có:

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

\(\Rightarrow2A=2\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

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

\(\Rightarrow2A-A=2.2^2+\left(3.2^3-2.2^3\right)+...+\left(n-n+1\right).2^n-n.2^{n+1}\)

\(\Rightarrow A=2.2^2+2^3+2^4+...+2^n-n.2^{n+1}\)

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

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

Đặt \(B=2^2+2^3+...+2^{n+1}\)

\(\Rightarrow2B=2^3+2^4+...+2^{n+2}\)

\(\Rightarrow2B-B=2^{n+2}-2^2\Rightarrow B=2^{n+2}-2^2\)

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

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

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

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

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

\(\Rightarrow2\left(n+1\right)=2^{10}\Rightarrow n-1=2^9\)

\(\Rightarrow n-1=512\Rightarrow n=513\)

Vậy \(n=513\)