The ratio of wavelengths $({\mathrm{\lambda }}_2/{\mathrm{\lambda }}_1)$  of first two lines of Balmer series $({\mathrm{\lambda }}_2>{\mathrm{\lambda }}_1)$ is:

Rydberg equation 

$\overrightarrow{\mathrm{\nu }}=\frac{1}{\mathrm{\lambda }}={\mathrm{R}}_{\mathrm{H}}\left[\frac{{\displaystyle 1}}{{\displaystyle {\mathrm{n}}_1^2}}-\frac{{\displaystyle 1}}{{\displaystyle {\mathrm{n}}_2^2}}\right]$

For Balmer series ${\mathrm{n}}_1=2,{\mathrm{n}}_2=3,4,5,....$

$\overrightarrow{{\mathrm{\nu }}_1}=\frac{1}{{\mathrm{\lambda }}_1}={\mathrm{R}}_{\mathrm{H}}\left[\frac{{\displaystyle 1}}{{\displaystyle 2^2}}-\frac{{\displaystyle 1}}{{\displaystyle 3^2}}\right]=\frac{5{\mathrm{R}}_{\mathrm{H}}}{36}$

$\overrightarrow{{\mathrm{\nu }}_2}=\frac{1}{{\mathrm{\lambda }}_2}={\mathrm{R}}_{\mathrm{H}}\left[\frac{{\displaystyle 1}}{{\displaystyle 2^2}}-\frac{{\displaystyle 1}}{{\displaystyle 4^2}}\right]=\frac{(4-1){\mathrm{R}}_{\mathrm{H}}}{16}=\frac{3{\mathrm{R}}_{\mathrm{H}}}{16}$

$\frac{\overrightarrow{{\mathrm{\nu }}_1}}{\overrightarrow{{\mathrm{\nu }}_2}}=\frac{1}{{\mathrm{\lambda }}_1}\times \frac{{\mathrm{\lambda }}_2}{1}=\frac{5{\mathrm{R}}_{\mathrm{H}}}{36}\times \frac{16}{3{\mathrm{R}}_{\mathrm{H}}}=\frac{20}{27}$