The ratio of wavelengths (λ2/λ1) of first two lines of Balmer series (λ2>λ1) is:

Rydberg equation

$\vec{ν} = \frac{1}{λ} = R_{H} [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}]$

For Balmer series $n_{1} = 2, n_{2} = 3, 4, 5, . ...$

$\vec{ν_{1}} = \frac{1}{λ_{1}} = R_{H} [\frac{1}{2^{2}} - \frac{1}{3^{2}}] = \frac{5 R_{H}}{36}$

$\vec{ν_{2}} = \frac{1}{λ_{2}} = R_{H} [\frac{1}{2^{2}} - \frac{1}{4^{2}}] = \frac{(4 - 1) R_{H}}{16} = \frac{3 R_{H}}{16}$

$\frac{\vec{ν_{1}}}{\vec{ν_{2}}} = \frac{1}{λ_{1}} \times \frac{λ_{2}}{1} = \frac{5 R_{H}}{36} \times \frac{16}{3 R_{H}} = \frac{20}{27}$

The ratio of wavelengths $(λ_{2} / λ_{1})$ of first two lines of Balmer series $(λ_{2} > λ_{1})$ is: