For a given series of spectral lines of atomic hydrogen : Let $\mathrm{\Delta v}={\mathrm{v}}_{max}-{\mathrm{v}}_{min}$ be the difference in maximum and minimum frequencies in 

cm^-1. Then ratio of ${\mathrm{\Delta V}}_{{ }_{\text{Lyman }}}/{\mathrm{\Delta V}}_{{ }_{\text{Balmer }}}\text{ is : }$

$\begin{array}{l}{\mathrm{v}}_{{\mathrm{L}}_{min}}={\mathrm{R}}_{\mathrm{H}}\left[\frac{1}{1^2}-\frac{1}{2^2}\right]=\frac{3}{4}{\mathrm{R}}_{\mathrm{H}}\\ {\mathrm{v}}_{{\mathrm{L}}_{max}}={\mathrm{R}}_{\mathrm{H}}\left[\frac{1}{1^2}-\frac{1}{{\mathrm{\infty }}^2}\right]={\mathrm{R}}_{\mathrm{H}}\\ \therefore {\left[{\mathrm{v}}_{max}-{\mathrm{v}}_{min}\right]}_{{\mathrm{Ly}}_{\mathrm{man}}}={\mathrm{R}}_{\mathrm{H}}-\frac{3}{4}{\mathrm{R}}_{\mathrm{H}}=\frac{{\mathrm{R}}_{\mathrm{H}}}{4}\dots \text{ (i) }\\ {\mathrm{v}}_{{\mathrm{B}}_{min}}={\mathrm{R}}_{\mathrm{H}}\left[\frac{1}{2^2}-\frac{1}{3^2}\right]=\frac{5}{36}{\mathrm{R}}_{\mathrm{H}}\\ {\mathrm{v}}_{\mathrm{B}}max={\mathrm{R}}_{\mathrm{H}}\left[\frac{1}{2^2}-\frac{1}{{\mathrm{\infty }}^2}\right]=\frac{1}{4}{\mathrm{R}}_{\mathrm{H}}\\ \therefore \left[{\mathrm{v}}_{max}\right.{\left.-{\mathrm{v}}_{min}\right]}_{\text{Balmer }}=\frac{{\mathrm{R}}_{\mathrm{H}}}{4}-\frac{5{\mathrm{R}}_{\mathrm{H}}}{36}=\frac{4{\mathrm{R}}_{\mathrm{H}}}{36}=\frac{{\mathrm{R}}_{\mathrm{H}}}{9}\dots \text{ (ii) }\\ \text{ By (i) /(ii) }\frac{{\mathrm{\Delta v}}_{\mathrm{L}}}{{\mathrm{\Delta v}}_{\mathrm{B}}}=\frac{9}{4}\end{array}$