As we know, wavelength of the light emitted during transition is given by,

$\frac{1}{\lambda }=R\left[\frac{1}{{\displaystyle {\left(n_1\right)}^2}}-\frac{{\displaystyle 1}}{{\displaystyle {\left(n_2\right)}^2}}\right]$.

Frequency can be written as,

$\nu =cR\left[\frac{1}{{\displaystyle {\left(n_1\right)}^2}}-\frac{{\displaystyle 1}}{{\displaystyle {\left(n_2\right)}^2}}\right]$

Now, for Balmer series$\left(2 \text{to} \infty \right)$,

${\nu }_B=cR\left[\frac{1}{{\displaystyle {\left(2\right)}^2}}-\frac{{\displaystyle 1}}{{\displaystyle {\left(\infty \right)}^2}}\right] ....Eq\left(1\right)$

For Brackett series we can write$\left(4 \text{to} \infty \right)$,

${\nu }^{\text{'}}=cR\left[\frac{1}{{\displaystyle {\left(4\right)}^2}}-\frac{{\displaystyle 1}}{{\displaystyle {\left(\infty \right)}^2}}\right] ....Eq\left(2\right)$

From equation 2 and equation 1, we get

$$\begin{gathered} \frac{{\nu }^{\text{'}}}{{\nu }_B}=\frac{1}{4} \\ \Rightarrow {\nu }^{\text{'}}=\frac{{\nu }_B}{4} \end{gathered}$$