Question

How many terms of the GP $\frac{8}{9},\frac{8}{18},\frac{8}{36}$..... should be added to get the sum $\frac{31}{18}$?

Answer 1

Here, $a=\frac{8}{9},r=\frac{\frac{8}{18}}{\frac{8}{9}}=\frac{1}{2}$ and $S_n=\frac{31}{18}.$
Since $Sn=\frac{a(1-r^n)}{1-r}$ is the sum of $n$ terms of a GP of first term $a$ and common ratio $r,$ we have
$\frac{31}{18}=\frac{\frac{8}{9}\times (1-{\left(\frac{1}{2}\right)}^n)}{1-\left(\frac{1}{2}\right)}$

$\Rightarrow$ $\frac{31}{36}=\frac{8}{9}\times (1-{\left(\frac{1}{2}\right)}^n)$

$\Rightarrow \frac{31}{32}=1-(\frac{1}{2}{)}^n$

$\Rightarrow 1-\frac{31}{32}=(\frac{1}{2}{)}^n$
$\Rightarrow \frac{1}{32}=(\frac{1}{2}{)}^n$
i.e., $\frac{1}{2^5}=\frac{1}{2^n}$
$\Rightarrow n=5$