Question

Sum of certain number of terms of the series $\frac{2}{9},-\frac{1}{3},\frac{1}{2},...$ is $\frac{55}{72}$. Find the number.

Answer 1

Here $a=\frac{2}{9}$ and $r=\frac{\frac{1}{3}}{\frac{9}{2}}=-\frac{3}{2}$. Let the number of terms be $n$.

Also, $S_n=\frac{a(1-r^n)}{1-r}=\frac{55}{72}$
$\therefore \frac{2}{9}\cdot \frac{[1-(-3/2{)}^n]}{1-(-3/2)}=\frac{55}{72}$

$\Rightarrow 1-{(-\frac{3}{2})}^n=\frac{55}{72}\times \frac{9}{2}\times \frac{5}{2}=\frac{275}{32}$
$\Rightarrow 1-\frac{275}{32}={(-\frac{3}{2})}^n$

$or-\frac{243}{32}={(-\frac{3}{2})}^{n}or{(-\frac{3}{2})}^5={(-\frac{3}{2})}^n\therefore n=5$