Question

\(\frac{3}{9\cdot15}+\frac{3}{15\cdot21}+\frac{3}{21\cdot27}+.....+\frac{3}{99\cdot105}\)

Answer 1

=\(\frac{1}{2}\cdot\left(\frac{6}{9\cdot15}+\frac{6}{15\cdot21}+\frac{6}{21\cdot27}+...+\frac{6}{99\cdot105}\right)\)

=\(\frac{1}{2}\cdot\left(\frac{1}{9}-\frac{1}{15}+\frac{1}{15}-\frac{1}{21}+\frac{1}{21}-\frac{1}{27}+...+\frac{1}{99}-\frac{1}{105}\right)\)

=\(\frac{1}{2}\cdot\left(\frac{1}{9}-\frac{1}{105}\right)\)

=\(\frac{1}{2}\cdot\left(\frac{35}{315}-\frac{3}{315}\right)\)

=\(\frac{1}{2}\cdot\left(\frac{32}{315}\right)\)

=\(\frac{1\cdot16}{1\cdot315}\)

=\(\frac{16}{315}\)