Question

Find the sum $\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{9\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}$.

Answer 1

$\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{9\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}$.

$=\frac{7\times 9+5\times 7}{5\times 7\times 7\times 9}+\frac{11\times 13+9\times 11}{9\times 11\times 11\times 13}+\frac{1}{13\times 15}$

$=\frac{14}{5\times 7\times 9}+\frac{22}{9\times 11\times 13}+\frac{1}{13\times 15}$

$=\frac{2}{5\times 9}+\frac{2}{9\times 13}+\frac{1}{13\times 15}$

$=\frac{2\times 9\times 13+2\times 5\times 9}{5\times 9\times 9\times 13}+\frac{1}{13\times 15}$

$=\frac{2\times 13+2\times 5}{5\times 9\times 13}+\frac{1}{13\times 15}$

$=\frac{26+10}{5\times 9\times 13}+\frac{1}{13\times 15}$

$=\frac{36}{5\times 9\times 13}+\frac{1}{13\times 15}$

$=\frac{4}{5\times 13}+\frac{1}{13\times 15}$

$=\frac{4\times 13\times 15+5\times 13}{5\times 13\times 13\times 15}$

$=\frac{4\times 15+5}{5\times 13\times 15}$

$=\frac{65}{5\times 13\times 15}$

$=\frac{1}{15}$