For a hydrogenic ion kinetic energy of electron in its 3rd excited state is found to be 54.4 eV. Then series limit $\left(\frac{1}{\lambda }\right)$ for Balmer series, for this ion, is

$\begin{array}{l}\mathrm{KE}=13.6\frac{{\mathrm{Z}}^2}{{\mathrm{n}}^2}\mathrm{eV}/\mathrm{atom}\\ \Rightarrow 54.4=13.6\times \frac{{\mathrm{Z}}^2}{4^2}\\ \Rightarrow {\mathrm{Z}}^2=64\\ \mathrm{for}\mathrm{Balmer}\mathrm{series}\\ \frac{1}{\mathrm{\lambda }}={\mathrm{RZ}}^2\left[\frac{1}{2^2}-\frac{1}{{\mathrm{\infty }}^2}\right]\left(\mathrm{for}\mathrm{series}\mathrm{limit}\right)\\ =109678\times 64\left[\frac{1}{4}-\frac{1}{\mathrm{\infty }}\right]{\mathrm{cm}}^{-1}\\ =109678\times 16{\mathrm{cm}}^{-1}\end{array}$