Giải:
Đặt \(S=2.2^2+3.2^3+...+n.2^n=2^{n+11}\)
\(S=2S-S=\left(2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\right)-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)
\(S=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^{n-1}+2^n\right)\)
Đặt \(T=2^3+2^4+...+2^{n-1}+2^n\)
Ta tính được: \(T=2T-T=2^{n-1}-2^3\)
\(\Rightarrow S=n.2^{n+1}-2^3-2^{n-1}+2^3=\left(n-1\right).2^{n+1}\)
\(\Rightarrow\left(n-1\right).2^{n+1}=n^{n+11}\)
\(\Rightarrow n-1=2^{10}\)
\(\Rightarrow n=2^{10}+1\)
\(n=1024+1\)
\(\Rightarrow n=1025\)
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