The ratio of specific heats $\left(\frac{C_P}{C_V}\right)$ in terms of degree of freedom (f) is given by:

Molar heat capacity at constant volume ${\mathrm{C}}_{\mathrm{V}}=\frac{\mathrm{fR}}{2}$ where f is degree of freedom.

Molar heat capacity at constant pressure can be written as ${\mathrm{C}}_{\mathrm{P}}=\mathrm{R}+{\mathrm{C}}_{\mathrm{V}}=\mathrm{R}+\frac{\mathrm{fR}}{2}=\left(1+\frac{\mathrm{f}}{2}\right)\mathrm{R}$
So $\frac{{\mathrm{C}}_{\mathrm{p}}}{{\mathrm{C}}_{\mathrm{v}}}=1+\frac{2}{\mathrm{f}}$