Question

Sum $1,\sqrt{3},3,....$ to $12$ terms.

Answer 1

Clearly the given series is a G.P. with common ratio $r=\sqrt{3}$ and first term $a=1$
$S_n$ be the sum of first $n$ terms of G.P.
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
$n=12$ ....... (given)

Then, $S_{12}=\frac{1\times \left(\right(\sqrt{3}{)}^{12}-1)}{\sqrt{3}-1}$
$\Rightarrow S_{12}=\frac{1\times \left(\right(3{)}^6-1)}{\sqrt{3}-1}$

$\Rightarrow S_{12}=\frac{(3^6-1)(\sqrt{3}+1)}{2}$

$\Rightarrow S_{12}=364(\sqrt{3}+1)$