Question

Tìm n sao cho 2.2² + 3.2³ + ............+ n.2ⁿ = 2ⁿ⁺¹¹

Answer 1

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

\(\Rightarrow2A=2\left(2.2^2+3.2^3+...+n.2^n\right)\)

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

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

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

Đặt \(B=2^3+2^4+2^5+...+2^{n-1}+2^n\)

Ta tính được \(B=2B-B=2^{n-1}-2^3\)

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

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

\(\Rightarrow n-1=2^{10}\Rightarrow n=2^{10}+1=1025\)

Vậy \(n=1025\)