Molar mass $$M - {{2 \times 4 + n \times 1} \over {2 + n}}$$ ..... (i)

Also, $$\gamma = {{{n_1}{C_{{P_1}}} + {n_2}{C_{{P_2}}}} \over {{n_1}{C_{{V_1}}} + {n_2}{C_{{V_2}}}}} = {{2 \times 5R + n \times 7R} \over {2 \times 3R + n \times 5R}}$$

$$ \Rightarrow \gamma = {{10 + 7n} \over {6 + 5n}}$$ ...... (ii)

Given that $${V_{rms}} = \sqrt 2 \,{V_{sound}}$$

$$ \Rightarrow \sqrt {{{3RT} \over M}} = \sqrt 2 \sqrt {{{\gamma RT} \over M}} $$

$$ \Rightarrow \gamma = {3 \over 2}$$

$$ \Rightarrow n = 2$$