Explanation:
For Lyman series
\(v = Rc\left[ {\frac{1}{{{1^2}}} - \frac{1}{{{n^2}}}} \right]\)
where \(n = 2,3,4.....\)
For the series limit of Lyman series \(n = \infty \)
\(\therefore {v_1} = Rc\left[ {\frac{1}{{{1^2}}} - \frac{1}{{{\infty ^2}}}} \right] = Rc\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
For the first line of Lyman series, \(n = 2\)
\(\therefore {v_2} = Rc\left[ {\frac{1}{{{1^2}}} - \frac{1}{{{2^2}}}} \right] = \frac{3}{4}Rc\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
For Balmer series
\(v = Rc\left[ {\frac{1}{{{2^2}}} - \frac{1}{{{n^2}}}} \right]\)
where \(n = 3,4,5....\)
For the series limit of Balmer series \(n = \infty \)
\(\therefore {v_3} = Rc\left[ {\frac{1}{{{2^2}}} - \frac{1}{{{\infty ^2}}}} \right] = \frac{{Rc}}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)\)
From equations (1), (2) and (3), we get
\({v_1} = {v_2} + {v_3}\) or \({v_1} - {v_2} = {v_3}\)