Question

Find the sum to $n$ terms of a G.P. $\sqrt{7},\sqrt{21},3\sqrt{7}...$

Answer 1

Given, $\sqrt{7},\sqrt{21},3\sqrt{7}...$
First term $=a=\sqrt{7}$ and
Common ratio $=r=\frac{\sqrt{21}}{\sqrt{7}}=\frac{\sqrt{7}\times \sqrt{3}}{\sqrt{7}}=\sqrt{3}>1$

We know, the sum of $n$ terms of $G.P.$ is
$S_n=\frac{a(r^n-1)}{r-1}$
$\Rightarrow S_n=\frac{\sqrt{7}({\sqrt{3}}^n-1)}{\sqrt{3}-1}$
By using rationalization,
$S_n=\frac{\sqrt{7}\left(\right(\sqrt{3}{)}^n-1)}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}$

$\Rightarrow S_n=\frac{\sqrt{7}\left(\right(3{)}^{\frac{n}{2}}-1)}{3-1}\times (\sqrt{3}+1)$

$\Rightarrow S_n=\frac{\sqrt{7}\left(\right(3{)}^{\frac{n}{2}}-1)}{2}\times (\sqrt{3}+1)$

$\therefore S_n=\frac{\sqrt{7}(\sqrt{3}+1)\left(\right(3{)}^{\frac{n}{2}}-1)}{2}$