Question

Rút gọn \(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{2012+\frac{2012}{2}+\frac{2011}{3}+...+\frac{1}{2013}}\)

Answer 1

\(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{2012+\frac{2012}{2}+\frac{2011}{3}+...+\frac{1}{2013}}\)

\(=\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{\left(\frac{2012}{2}+1\right)+\left(\frac{2011}{3}+1\right)+....+\left(\frac{1}{2013}+1\right)}\)

\(=\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2003}}{\frac{2014}{2}+\frac{2014}{3}+...+\frac{2014}{2013}}\)

\(=\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2003}}{2014\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2003}\right)}\)

\(=\frac{1}{2014}\)