Question

Tím số tự nhiên n thỏa mãn 

2*2^2+3*2^3+4*2^4+.....+(n-1)*2^n-1+n*2^n=2^n+31

Answer 1

Đặt \(A=2\cdot2^2+3\cdot2^3+.....+n\cdot2^n\)

\(\Rightarrow2A=2\cdot2^3+3\cdot2^4+....+n\cdot2^{n+1}\)

\(\Rightarrow2A-A=\left(2\cdot2^3-3\cdot2^3\right)+\left(3\cdot2^4-4\cdot2^4\right)+....+\left(\left(n-1\right)2^n-n\cdot2^n\right)+n\cdot2^{n+1}-2^3\)

\(\Rightarrow A=n\cdot2^{n+1}-2^3-\left(2^3+2^4+2^5+....+2^n\right)\)

Đặt \(B=2^3+2^4+...+2^n\)

\(\Rightarrow2B=2^4+2^5+...+2^{n+1}\)

\(\Rightarrow B=2^{n+1}-2^3\)

\(\Rightarrow A=n\cdot2^{n+1}-2^3-\left(2^{n+1}-2^3\right)\)

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

\(\Rightarrow2^{n+1}\left(n-1\right)=2^{n+31}\)

\(\Rightarrow n-1=2^{30}\)

\(\Rightarrow n=2^{30}+1\)