Question

\(\frac{x-1}{2010}+\frac{x-2}{2009}+.......+\frac{x-2010}{1}=2010cacbangiupminhhen\)

Answer 1

\(\frac{x-1}{2010}+\frac{x-2}{2009}+....+\frac{x-2010}{2}=2010\)

\(\Leftrightarrow\left(\frac{x-1}{2010}-1\right)+\left(\frac{x-2}{2009}-1\right)+...+\left(\frac{x-2010}{1}-1\right)=0\)

\(\Leftrightarrow\frac{x-2011}{2010}+\frac{x-2011}{2009}+....+\frac{x-2011}{1}=0\)

\(\Leftrightarrow\left(x-2011\right)\left(\frac{1}{2010}+\frac{1}{2009}+...+1\right)=0\)

\(\Rightarrow x-2011=0\Rightarrow x=2011\)