\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+....+\frac{2}{x.\left(x+1\right)}=\frac{1999}{2001}\)
\(\Leftrightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x.\left(x+1\right)}=\frac{1999}{2001}\)
\(\Leftrightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+....+\frac{2}{x.\left(x+1\right)}=\frac{1999}{2001}\)
\(\Leftrightarrow2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{199}{2001}\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{1999}{2001}\div2\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{1999}{4002}\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{1999}{4002}\Leftrightarrow\frac{1}{x+1}=\frac{1}{2001}\)
\(\Leftrightarrow x+1=2001\Rightarrow x=2000\)
