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)

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

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

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