\(S=\frac{5}{2\cdot3\cdot4}+\frac{5}{3\cdot4\cdot5}+......+\frac{5}{99\cdot100\cdot101}\)
\(S\frac{2}{5}=\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+.....+\frac{2}{99\cdot100\cdot101}\)
\(\frac{2}{2\cdot3\cdot4}=\frac{1}{2\cdot3}-\frac{1}{3\cdot4}\)
\(\frac{2}{3\cdot4\cdot5}=\frac{1}{3\cdot4}-\frac{1}{4\cdot5}\)
.............
\(\frac{2}{99\cdot100\cdot101}=\frac{1}{99\cdot100}-\frac{1}{100\cdot101}\)
\(\Rightarrow S\frac{2}{5}=\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+.........+\frac{1}{99\cdot100}-\frac{1}{100\cdot101}\)
\(\Rightarrow S\frac{2}{5}=\frac{1}{2\cdot3}-\frac{1}{100\cdot101}\)
\(\Rightarrow S\frac{2}{5}=\frac{1}{6}-\frac{1}{10100}\)
\(\Rightarrow S\frac{2}{5}=\frac{5047}{30300}\)
\(\Rightarrow S=\frac{5047}{30300}:\frac{2}{5}\)
\(\Rightarrow S=\frac{5047}{30300}\cdot\frac{5}{2}\)
\(\Rightarrow S=\frac{5047}{12120}\)