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# Find the sum upto $n$ terms of the G.P. $\sqrt 7,\sqrt {21},3\sqrt 7,.......$

$\begin{array}{1 1}\sqrt 7.(\sqrt 3^n-1)(\sqrt 3+1) \\ \frac{\sqrt 7}{2}.(\sqrt 3^n-1)(\sqrt 3-1) \\\frac{\sqrt 7}{2}.(3^n-1) \\ \frac{\sqrt 7}{2}.(\sqrt 3^n-1)(\sqrt 3+1) \end{array}$

Toolbox:
• Sum of $n$ terms of a G.P.=$S_n=a.\large\frac{r^n-1}{r-1}$ where $a=$first term and $r=$ common ratio.
Given sequence is $\sqrt 7,\sqrt {21},3\sqrt 7........$
In this G.P. first term $=a=\sqrt 7$ and
common ratio$=\large\frac{\sqrt {21}}{\sqrt 7}=\frac{\sqrt 3\times\sqrt 7}{\sqrt 7}$$=\sqrt 3 We know that the sum of n terms of a G.P.=S_n=a.\large\frac{r^n-1}{r-1} \therefore Sum of n terms of the given sequence =\sqrt 7.\large\frac{\sqrt 3^n-1}{\sqrt 3-1} Rationalising the denominator we get S_n=\sqrt 7.\large\frac{\sqrt 3^n-1}{\sqrt 3-1}.\frac{\sqrt 3+1}{\sqrt 3+1}=\frac{\sqrt 7}{2}.$$(\sqrt 3^n-1)(\sqrt 3+1)$