$v _{ mms }=\sqrt{\frac{3 RT }{ M }}$

$\frac{ v _{ s }}{ v _{ rms }}=\sqrt{\frac{\gamma}{3}}=\frac{1}{\sqrt{2}} \Rightarrow \frac{\gamma}{3}=\frac{1}{2} \Rightarrow \gamma=\frac{3}{2}$

$\gamma=1+\frac{2}{ f _{\operatorname{mix}}}$

$f _{\operatorname{mix}}=\frac{2 \times 3+ n \times 5}{ n +2}=\frac{6+ n \times 5}{( n +2)}$

$\gamma=1+\frac{2( n +2)}{6+ n \times 5}=\frac{6+5 n +2 n +4}{6+5 n }$

$\gamma=\frac{7 n +10}{6+5 n }=\frac{3}{2}$

$14 n +20=18+15 n$

$n =2$