If $v_1$ is the frequency of the series limit of Lyman series, $v_2$ is the frequency of the first line of Lyman series and $v_3$ is the frequency of the series limit of the Balmer series, then:

For Lyman series, $v_{\mathrm{L}}=\mathrm{Rc}\left[\frac{1}{1^2}-\frac{1}{{\mathrm{n}}^2}\right]$

where $n=2,3,4,.......$

For the series limit of Lyman series, $n=\infty$

$\therefore v_1=\mathrm{Rc}\left[\frac{1}{1^2}-\frac{1}{{\infty }^2}\right]=\mathrm{Rc} .....\left(i\right)$

For the first line of Lyman series, $n=2$

$\therefore v_2=\mathrm{Rc}\left[\frac{1}{1^2}-\frac{1}{2^2}\right]=\frac{3}{4}\mathrm{Rc} .....\left(ii\right)$

For Balmer series 

$v_{\mathrm{B}}=\mathrm{Rc}\left[\frac{1}{2^2}-\frac{1}{{\mathrm{n}}^2}\right]$

where $n= 3,4,5,....$

For the series limit of Balmer series, $n=\infty$

$\therefore v_3=\mathrm{Rc}\left[\frac{1}{2^2}-\frac{1}{{\infty }^2}\right]=\frac{\mathrm{Rc} }{4} .....\left(iii\right)$

From equations (i), (ii) and (iii), we get

$v_1=v_2+v_3 or, v_1-v_2=v_3$