Question

[nguyễn tuấn anh]

Tìm x biết x=(2009:2)+(2008:2²)+(2007:2³)+...+(1:2²⁰⁰⁹)

Answer 1

[Akai Haruma]

Lời giải:

\(x=\frac{1}{2^{2009}}+\frac{2}{2^{2008}}+\frac{3}{2^{2007}}+....+\frac{2008}{2^2}+\frac{2009}{2}\)

\(2x = \frac{1}{2^{2008}}+\frac{2}{2^{2007}}+\frac{3}{2^{2006}}+...+\frac{2008}{2}+2009\)

\(\Rightarrow x=2x-x=2009-\frac{1}{2}-\frac{1}{2^2}-...-\frac{1}{2^{2008}}-\frac{1}{2^{2009}}\)

\(\Rightarrow 2009-x=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2008}}+\frac{1}{2^{2009}}\)

\(\Rightarrow 2(2009-x)=1+\frac{1}{2}+....+\frac{1}{2^{2007}}+\frac{1}{2^{2008}}\)

\(\Rightarrow 2(2009-x)-(2009-x)=1-\frac{1}{2^{2009}}\)

\(\Rightarrow 2009-x=1-\frac{1}{2^{2009}}\\ \Rightarrow x=2009-(1-\frac{1}{2^{2009}})=2008+\frac{1}{2^{2009}}\)