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

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

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

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

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

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

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

Từ đây phương trình ban đầu tương đương với: 

\(\left(2n-2\right).2^n=2^{n+34}\)

\(\Leftrightarrow\left(2n-2\right).2^n=2^n.2^{34}\)

\(\Leftrightarrow n-1=2^{33}\)

\(\Leftrightarrow n=2^{33}+1\)

Tham khảo : Tìm số tự nhiên n thoả mãn 2.2^2 +3.2^3 +4.2^4+ ...+n.2^n = 2^n+11 - My Hien

Tham khảo : Tìm số tự nhiên n thoả mãn - My Hien

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

\(\Leftrightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)

\(\Leftrightarrow2A-A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

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

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

mà \(A=2^{n+11}\) \(\Leftrightarrow\left(n-1\right).2^{n+1}=2^{n+11}\)

\(\Leftrightarrow\left(n-1\right).2^n.2=2^n.2^{11}\)

\(\Leftrightarrow\left(n-1\right)=2^{10}\)

\(\Leftrightarrow n=2^{10}+1\)