The wavelength of $H_{\alpha }$ line of Balmer series is $\mathrm{X}\stackrel{o}{A}.$ What is the wave length of $H_{\beta }$ line of Balmer series in terms of $\text{'}\mathrm{X}\text{'}$

For Balmer series, $\frac{1}{\lambda }=R[\frac{1}{2^2}-\frac{1}{n_2^2}]$

For alpha line, $\frac{1}{{\lambda }_{\alpha }}=R[\frac{1}{2^2}-\frac{1}{3^2}]=\frac{1}{X}$

$\frac{1}{{\lambda }_{\alpha }}=R\times \frac{5}{36}=\frac{1}{X}$

$R=\frac{36}{5}\times \frac{1}{X}\mathrm{....}\left(1\right)$

For beeta line, $\frac{1}{{\lambda }_{\beta }}=R[\frac{1}{2^2}-\frac{1}{4^2}]=\frac{3}{16}R\mathrm{............}\left(2\right)$

Substitute eq $\left(1\right)$ into $\left(2\right)$

$\frac{1}{{\lambda }_{\beta }}=\frac{3}{16}R=\frac{3}{16}\times \frac{36}{5}\times \frac{1}{X}=\frac{108}{80}\times \frac{1}{X}$

${\lambda }_{\beta }=X\frac{80}{108}{A}^o.$