\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{x.\left(x+1\right)}=\frac{2005}{2010}\)
\(\Leftrightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{401}{402}\)
\(\Leftrightarrow1-\frac{1}{x+1}=\frac{401}{402}\)
\(\Leftrightarrow\frac{1}{x+1}=1-\frac{401}{402}\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{402}\)
\(\Leftrightarrow x+1=402\Rightarrow x=401\)