# How many terms of the series $\sqrt2$, 2, $2\sqrt2$,------- will make a sum of $62+31\sqrt2$ ?

$\begin{array}{1 1} 11 \\ 21 \\ 10 \\ 20 \end{array}$

Explanation : $\sqrt2,2,2\sqrt2$ is a GP with $a\sqrt2\quad\;r=\sqrt2$
$S_{n}=\large\frac{\sqrt2((\sqrt2)^n-1)}{\sqrt2-1}=62+31\sqrt2$
$\sqrt2 \;(\sqrt2^n-1)=(62+31\sqrt2)(\sqrt2-1)$
$\sqrt2 \;(\sqrt2^n-1)=62\;\sqrt2-62+62-31\sqrt2$
$\sqrt2 \;(\sqrt2^n-1)=31\sqrt2$
$\sqrt2^n-1=31$
$\sqrt2^n=32$
$n=10.$
edited Jan 24, 2014