Question

Of how many terms is $\frac{55}{72}$, the sum of the series $\frac{2}{9}-\frac{1}{3}+\frac{1}{2}-....?$

Answer 1

This is G.P series whose first term is $a$ and $r$ is common ratio

$\begin{array}{c}r=\frac{-1}{3}/\frac{2}{3}\\ =\frac{-3}{2}\end{array}$

$\Rightarrow \frac{55}{72}=\frac{\left[\frac{2}{9}\left(1-{\left(\frac{-3}{2}\right)}^n\right)\right]}{1-\left(\frac{-3}{2}\right)}$

by using formula $S_n=\frac{a\left(1-r^n\right)}{1-r}$

on solving this we get

$n=5$