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 \(n=513\)
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