Let us consider 1 mole of an ideal gas at kelvin temperature $T .$ It has $N$ molecules (Avogadro's number). The internal energy of an ideal gas is entirely kinetic.
The average KE per molecule of a ideal gas is $\frac{1}{2} n k T$ (k is boltzman constant), where $n$ is degree of freedom. Therefore the internal energy of one mole of an gas would be
$E=N\left(\frac{1}{2} n k T\right)=\frac{1}{2} n R T$
$\left(\because k=\frac{R}{N}\right)$
Now, $C_{V}=\frac{d E}{d T}=\frac{n}{2} R$
and $C_{p}=\frac{n}{2} R+R=\left(\frac{n}{2}+1\right) R$
$\frac{C_{p}}{C_{V}}=\frac{\left(\frac{n}{2}+1\right) R}{\frac{n}{2}}$
$=\left(1+\frac{2}{n}\right)$