Question

Find Sun of the below series
1/(1*2) + 1/(2*3) + 1/(3*4) + ...... +1/(12*13) +1/(13*14)

Answer 1

\[\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}+ ... + \frac{1}{182}\]\[= \frac{1}{2} + \left( \frac{1}{2} - \frac{1}{3}\right) +\left( \frac{1}{3} - \frac{1}{4}\right) + \left( \frac{1}{4} - \frac{1}{5}\right) + \left( \frac{1}{5} - \frac{1}{6}\right) + ...+\left( \frac{1}{13} - \frac{1}{14}\right) \]\[=\frac{1}{2} + \frac{1}{2} - \frac{1}{14}\]\[= \frac{13}{14}\]