Find the sum of the GP :


Find the sum of the GP :

$\sqrt{7}+\sqrt{21}+3 \sqrt{7}+\ldots$ to $\mathrm{n}$ terms



Sum of a G.P. series is represented by the formula $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{\mathrm{r}^{\mathrm{n}}-1}{\mathrm{r}-1}$ 

when r>1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.



$r=($ ratio between the $n$ term and $n-1$ term $) \sqrt{7} \div \sqrt{21}=\sqrt{3}$

n terms

$\therefore \mathrm{S}_{\mathrm{n}}=\sqrt{7} \times \frac{\sqrt{3}^{\mathrm{n}}-1}{\sqrt{3}-1}$ [Rationalizing the denominator]

$\Rightarrow S_{n}=\sqrt{7} \times \frac{\sqrt{3}^{n}-1}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1}$

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


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