Polytropic process

$P V^m=$ constant

Given heat capacities is average of $C_P$ and $C_V$ So

$C=\frac{C_P+C_V}{2}$

or $\quad C=\frac{2 C_V+R}{2}$

or $\quad C=\frac{C_V+R}{2} \quad \dots (i)$

Now formula for specific heat of polytropic process is given by

$C=\frac{R}{y-1}+\frac{R}{1-n} \quad \dots (ii)$

or $\frac{R}{y-1}+\frac{R}{2}=\frac{R}{y-1}+\frac{R}{1-n}$ as $C_V=\frac{R}{y-1}$

$\frac{R}{2}=\frac{R}{1-n}$

or $n=-1$