\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\times\left(x+1\right)}=\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\times\left(x+1\right)}=\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{2\times3}+\frac{2}{3\times4}+\frac{2}{4\times5}+...+\frac{2}{x\times\left(x+1\right)}=\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{2}-\frac{2}{3}+\frac{2}{3}-\frac{2}{4}+\frac{2}{4}-\frac{2}{5}+...+\frac{2}{x}-\frac{2}{x+1}=\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{2}-\frac{2}{x+1}=\frac{2005}{2007}\)
\(\Rightarrow1-\frac{2}{x+1}=\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{x+1}=1-\frac{2005}{2007}\)
\(\Rightarrow\frac{2}{x+1}=\frac{2}{2007}\)
\(\Rightarrow x+1=2007\)
\(\Rightarrow x=2006\)
\(\frac{1}{2}\cdot\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}\right)=\frac{1}{2}\cdot\frac{2005}{2007}\)
\(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{x\left(x+1\right)}=\frac{2005}{4014}\)
\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{2005}{4014}\)
\(\frac{1}{2}-\frac{1}{x+1}=\frac{2005}{4014}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2005}{4014}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2007}\)
\(\Rightarrow x+1=2007\)
\(x=2007-1\)
\(x=2006\)
