\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2000}{2002}\)
\(\Rightarrow\) \(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2000}{2002}\)
\(\Rightarrow\) \(2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2000}{2002}\)
\(\Rightarrow\) \(2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2000}{2002}\)
\(\Rightarrow\) \(2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2000}{2002}\)
\(\Rightarrow\) \(\frac{1}{2}-\frac{1}{x+1}=\frac{500}{1001}\)
\(\Rightarrow\) \(\frac{1}{x+1}=\frac{1}{2002}\)
\(\Rightarrow\) \(x+1=2002\) \(\Rightarrow\) \(x=2001\)
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x.\left(x+1\right)}=\frac{2000}{2002}\)
\(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x.\left(x+1\right)}\)=\(\frac{2000}{2002}\)
2.(\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x.\left(x+1\right)}\))=\(\frac{2000}{2002}\)
2.\(\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x.\left(x+1\right)}\right)=\frac{2000}{2002}\)
2.(\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}\)) = \(\frac{2000}{2002}\)
2.\(\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2000}{2002}\)
\(\frac{1}{2}-\frac{1}{x+1}=\frac{1000}{2002}\)
\(\frac{1}{x+1}=\frac{1}{2}-\frac{1000}{2002}\)
\(\frac{1}{x+1}=\frac{1}{2002}\)
2002.1 = (x+1).1
2002 = x+1
x=2001 (T/M)
