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Đặt \(A=2.2^2+3.2^3+4.2^4+5.2^5+...+n.2^n\)
\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)
\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)
\(-2.2^2-3.2^3-4.2^4-5.2^5-...-n.2^n\)
\(A=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^n\right)\)
Đặt \(M=\left(2^3+2^4+...+2^n\right)\)
\(\Rightarrow2M=\left(2^4+2^5+...+2^{n+1}\right)\)
\(\Rightarrow M=2^{n+1}-2^3\)
\(\Rightarrow A=n.2^{n+1}-2^3-2^{n+1}+2^3\)
\(\Rightarrow A=\left(n-1\right)2^{n+1}=2^{n+10}\)
\(\Rightarrow\left(n-1\right)=2^9\)
\(\Rightarrow n=513\)
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 \(A=2^{n+10}=2^{n+1}.2^9=2^{n+1}.512\)
\(\Rightarrow n-1=512\)
\(\Rightarrow n=513\)
Đặt A = 2.22 + 3.23 + 4.24 + ... + n.2n
2A = 2.23 + 3.24 + 4.25 + ... + n.2n+1
2A - A = (2.23 - 3.23) + (3.24 - 4.24) + ... + [(n-1).2n - n.2n] + n.2n+1
A = -23 - 24 - ... - 2n + n.2n+1 - 2.22
A = n.2n+1 - (23 + 24 + 25 + ... + 2n) - 23
Đặt B = 23 + 24 + ... + 2n
2B = 24 + 25 + ... + 2n+1
2B - B = 24 + 25 + ... + 2n+1 - 23 - 24 - 2n
B = 2n+1 - 23
Mà A = n.2n+1 - (23 + 24 + 25 + ... + 2n) - 23
=> A = n.2n+1 - B - 23
=> A = n.2n+1 - (2n+1 - 23) - 23
A = n.2n+1 - 2n+1 + 23 - 23
A = (n-1).2n+1
Mà 2.22+ 3.23 + 4.24 + 5.25 + · · · + n.2n = 2n+10
=> A = 2n+10
=> (n-1).2n+1 = 2n+10
(n-1) = 2n+10 : 2n+1
n-1 = 29
n = 512 + 1
n = 513
