\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2009}{2010}\)
\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2009}{2010}\)
\(\Rightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2009}{2010}\)
\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2009}{2010}\)
\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2009}{2010}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2009}{2010}:2\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2009}{4020}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2009}{4020}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{4020}\)
\(\Rightarrow x+1=4020\)
=> x = 4020 - 1
=> x = 4019
