\(C=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{120}\)
\(C=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{240}\)
\(C=\frac{2}{4\times5}+\frac{2}{5\times6}+\frac{2}{6\times7}+...+\frac{2}{15\times16}\)
\(C=2\times\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\right)\)
\(C=2\times\left(\frac{1}{4}-\frac{1}{16}\right)\)
\(C=2\times\frac{3}{16}=\frac{3}{8}\)
\(C=\frac{2}{20}+\frac{2}{30}+.........+\frac{2}{240}\)
\(=2\left(\frac{1}{4.5}+\frac{1}{5.6}+..........+\frac{1}{15.16}\right)\)
\(=2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+.......+\frac{1}{15}-\frac{1}{16}\right)\)
\(=2\left(\frac{1}{4}-\frac{1}{16}\right)\)
\(=2.\frac{3}{16}\)
\(=\frac{3}{8}\)