When plates of capacitor are separated by a dielectric medium of dielectric constant $K$, its capacity

$C_m=\frac{K{\epsilon }_{0}A}{d}=K{C}_0$

$ie, C_m=K{C}_0$

Here,$C_0=C$

$\therefore C_m=KC$

Now, two capacitors of capacities $KC$ and $C$ are in series, their effective capacitance

$\frac{1}{C^{\prime }}=\frac{1}{KC}+\frac{1}{C}$

or $\frac{1}{C^{\prime }} =\frac{1+K}{KC}$

$\therefore C^{\prime }=\frac{KC}{K+1}$