From quantization of angular momentum, one gets for hydrogen atom, the radius of the n^th orbit as ${\mathrm{r}}_{\mathrm{n}}=\frac{{\mathrm{n}}^2}{{\mathrm{m}}_{\mathrm{c}}}{\left(\frac{\mathrm{h}}{2\mathrm{\pi }}\right)}^2\left(\frac{4{\mathrm{\pi }}^2{\mathrm{\epsilon }}_0}{{\mathrm{e}}^2}\right)$. For a hydrogen like atom of atomic number ‘Z’,

For an atom with a single electron, Bohr atom model is applicable.
As the value of attraction between a proton and electron is proportional to e², for an ion with a single electron.

For an Ion with atomic number 'Z', $\frac{{\mathrm{e}}^2}{4{\mathrm{\pi \epsilon }}_0}$ is replaced by $\frac{{\mathrm{Ze}}^2}{4{\mathrm{\pi \epsilon }}_0}$.

Hence, radius becomes inversely proportional to Z.