\(\frac{1}{\lambda}=R\left[\frac{1}{1^{2}}-\frac{1}{2^{2}}\right]\)

The wavelength of the second line of Balmer series for hydrogen like ion is

\(\frac{1}{\lambda^{\prime}}=Z^{2} R\left[\frac{1}{2^{2}}-\frac{1}{4^{2}}\right]\)

According to question \(\lambda=\lambda^{\prime}\)

\(\Rightarrow R\left[\frac{1}{1^{2}}-\frac{1}{2^{2}}\right]=Z^{2} R\left[\frac{1}{2^{2}}-\frac{1}{4^{2}}\right]\)

or \(\frac{3}{4}=\frac{3 Z^{2}}{16} \quad\) or \(\quad Z^2=4\) or \(Z=2\)