Question

[pham hong quyen]

tinh tong 2*2²+3*2³+4*2⁴+...+ n*2ⁿ=2ⁿ⁺³⁴

Answer 1

[Yuri Sweet[𝕿𝖊𝖆𝖒 𝕹𝖊𝖕𝖆𝖑]]

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)

\(\Rightarrow2A-A=\left(2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\right)-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

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

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

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

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

\(\Rightarrow2^{n+1}\left(n-1\right)=2^{n+34}\)

\(\Rightarrow2^{n+1}\left(n-1\right)=2^{n+1}.2^{33}\)

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

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