When an atom comes down from some higher energy level to the first energy level then emitted lines form of Lyman series.

$\frac{1}{{\lambda }_L}$ =$R\left(\frac{1}{1^2}-\frac{1}{n^2}\right)$

where$R$ is Rydberg's constant.

When an atom comes from higher energy level to the second level, then Balmer series are obtained.

$\frac{1}{{\lambda }_B}$ =$R\left(\frac{1}{2^2}-\frac{1}{n^2}\right)$

For maximum wavelength

$n=2\text{,}\frac{1}{{\lambda }_L}$ =$R\left(1-\frac{1}{{\left(2\right)}^2}\right)$ = $R\left(1-\frac{1}{4}\right)=\frac{3R}{4} \dots . \left(i\right)$

$n=3$,$\frac{1}{{\lambda }_B}$ =$R\left(\frac{1}{{\left(2\right)}^2}-\frac{1}{{\left(3\right)}^2}\right)$ $= R\left(\frac{5}{36}\right) \dots \left(ii\right)$

Dividing Eq. (ii) by Eq. (i), we get

$\frac{ {\lambda }_L}{{\lambda }_B}=\frac{5}{27}$