\(=\frac{1}{2\cdot3:2}+\frac{1}{3\cdot4:2}+\frac{1}{4\cdot5:2}+...+\frac{1}{50\cdot51:2}\)
\(=\frac{2}{2\cdot3}+\frac{2}{3\cdot4}+\frac{2}{4\cdot5}+...+\frac{2}{100\cdot101}\)
\(=2\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{50\cdot51}\right)\)
\(=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{50}-\frac{1}{51}\right)\)
\(=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{51}\right)=\frac{1}{2}\cdot\frac{49}{102}=\frac{49}{204}\)