\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2007}{2009}\)
\(\Rightarrow\)\(2.\)(\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}=\frac{2007}{2009}\)
\(\Rightarrow\)\(2.\)(\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}=\frac{2007}{2009}\)
\(\Rightarrow\)\(2.\)(\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x\left(x+1\right)}\)) \(=\frac{2007}{2009}\)
\(\Rightarrow\)(\(\frac{1}{2}-\frac{1}{x+1}\))\(=\frac{2007}{2009}:2\)
\(\Rightarrow\frac{-1}{x+1}=\frac{2007}{4018}-\frac{1}{2}\)
\(\Rightarrow\frac{-1}{x+1}=\frac{-1}{4018}\Rightarrow x+1=4018\Rightarrow x=4017\)
