Question

[Bé Chanh]

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

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

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

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

\(\Leftrightarrow A=-2.2^2+\left(2.2^3-3.2^3\right)+\left(3.2^4-4.2^4\right)+...+[\left(n-1\right)2^n-n.2^n]+n.2^{n+1}\)

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

\(\Leftrightarrow A=-2^3-\left(2^4-2^3\right)-\left(2^5-2^4\right)-...-\left(2^{n+1}-2^n\right)+n.2^{n+1}\)

\(\Leftrightarrow A=-2^3-2^4+2^3-2^5+2^4-...-2^{n+1}+2^n+n.2^{n+1}\)

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

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

Mà \(A=2^{n+10}=2^{n+1}.2^9=2^{n+1}.512\)

\(\Rightarrow n-1=512\)

\(\Rightarrow n=513\)