Question

Sum of the series, $\sqrt{7},\sqrt{21},3\sqrt{7},...$ n terms.

Answer 1

The given G.P. is $\sqrt{7},\sqrt{21},3\sqrt{7},...$
Here $a=\sqrt{7},r=\frac{\sqrt{21}}{\sqrt{7}}=\sqrt{3}$
$\therefore S_n=\frac{a(1-r^n)}{1-r}=\frac{\sqrt{7}[1-(\sqrt{3}{)}^n]}{1-\sqrt{3}}=\frac{\sqrt{7}[1-(\sqrt{3}{)}^n]}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}}=\frac{\sqrt{7}(1+\sqrt{3})[1-({\sqrt{3}}^n]}{1-3}=\frac{\sqrt{7}(1+\sqrt{3})}{2}[1-({\sqrt{3}}^{\frac{n}{2}}]=\frac{\sqrt{7}(1+\sqrt{3})}{2}\left[\right(3{)}^{\frac{n}{2}}-1]$