# Find the sum to n terms in the geometric progression

Question:

Find the sum to n terms in the geometric progression $\sqrt{7}, \sqrt{21}, 3 \sqrt{7} \ldots$

Solution:

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

Here, $a=\sqrt{7}$

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

$\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{a}\left(1-\mathrm{r}^{\mathrm{n}}\right)}{1-\mathrm{r}}$

$\therefore S_{a}=\frac{\sqrt{7}\left[1-(\sqrt{3})^{n}\right]}{1-\sqrt{3}}$

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

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

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

$=\frac{\sqrt{7}(1+\sqrt{3})}{2}\left[(3)^{\frac{11}{2}}-1\right]$