Tính P = 1/2+!/3+!/4+..+1/2012/ 2011/1+ 2010/2+2009/3+...+1/2011

Tính $P=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}+\frac{1}{2012}}{\frac{2011}{1}+\frac{2010}{2}+\frac{2009}{3}+\mathrm{...}+\frac{1}{2011}}$

Trả lời

$\begin{array}{l}P=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}+\frac{1}{2012}}{\frac{2011}{1}+\frac{2010}{2}+\frac{2009}{3}+\mathrm{...}+\frac{1}{2011}}\\ P=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}+\frac{1}{2012}}{1+\left(1+\frac{2010}{2}\right)+\left(1+\frac{2009}{3}\right)+\mathrm{...}+\left(1+\frac{1}{2011}\right)}\\ P=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}+\frac{1}{2012}}{\frac{2012}{2012}+\frac{2012}{2}+\frac{2012}{3}+\mathrm{...}+\frac{2012}{2011}}\\ P=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}+\frac{1}{2012}}{2012\left(\frac{1}{2}+\frac{1}{3}+\mathrm{...}+\frac{1}{2011}+\frac{1}{2012}\right)}\\ P=\frac{1}{2012}\end{array}$