So with $v = n\lambda $, if $v$ increases $n$ will increase at

${27^o}C,\,{v_1} = n\lambda $, at ${31^o}C\,,\,\,{v_2} = (n + x)\lambda $

Now using $v \propto \sqrt T $         $(\because v= \sqrt{\frac{{\gamma \,R\,T}}{{M}}})$

$\frac{{{v_2}}}{{{v_1}}} = \sqrt {\frac{{{T_2}}}{{{T_1}}}} = \frac{{n + x}}{n}$

==> $\frac{{300 + x}}{{300}} = \sqrt {\frac{{(273 + 31)}}{{(273 + 27)}}} = \sqrt {\frac{{304}}{{300}}} = \sqrt {\frac{{300 + 4}}{{300}}} $

==> $1 + \frac{x}{{300}} = {\left( {1 + \frac{4}{{300}}} \right)^{1/2}} = \left( {1 + \frac{1}{2} \times \frac{4}{{300}}} \right)\,$                $[\because {(\,1 + x)^n} = 1 + nx]$

==> $x = 2.$