\(\frac{1}{3}+\frac{1}{6}=\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{499}{1000}\)
\(\frac{2}{6}+\frac{2}{12}+...+\frac{2}{x\left(x+1\right)}=\frac{499}{1000}\)
\(\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{x\left(x+1\right)}=\frac{499}{1000}\)
\(2\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{499}{1000}\)
\(2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{499}{1000}\)
\(2\left(\frac{1}{2}-\frac{1}{x-1}\right)=\frac{499}{1000}\)
\(\frac{1}{2}-\frac{1}{x-1}=\frac{499}{2000}\)
\(\frac{2000\left(x-1\right)}{2\left(x-1\right)2000}-\frac{2.2000}{\left(x-1\right)2.2000}=\frac{499.2\left(x-1\right)}{2000.2\left(x-1\right)}\)
Khử mẫu
\(2000x-2000-4000=998x-998\)
\(2000x-6000=998x-998\)
\(1002x-5002=0\)
\(1002x=5002\Leftrightarrow x=\frac{2501}{501}\)