Question

Find the sum to indicated number of terms in each of the geometric progressions in $\sqrt{7},\sqrt{21},3\sqrt{7},$ ... n terms.

Answer 1

The given G.P. is $\sqrt{7},\sqrt{21},3\sqrt{7},$ ...

Here, a = $\sqrt{7}$

r = $\frac{\sqrt{21}}{\sqrt{7}}=\sqrt{3}$

Sn = $\frac{a(1-r^n)}{1-r}$

∴ Sn = $\frac{\sqrt7[1-({\sqrt3})n]}{1-\sqrt3}$

= $\frac{\sqrt{7}[1-(\sqrt{3})n]}{1-\sqrt{3}}\times\frac{1+\sqrt{3}}{1+\sqrt{3}}$ (By rationalizing)

= $\frac{\sqrt{7}(1+\sqrt{3})[1-(\sqrt{3}n]}{1-3}$

= $-\frac{\sqrt{7}(1+\sqrt{3})}{2[\frac{1-(3)n}{2}]}$

= $\frac{\sqrt{7}(1+\sqrt{3})}{2\bigg[\frac{(3)n}{2}{-1}\bigg]}$