Explanation

(a) For Lyman series $v = RC\left\lbrack \frac{1}{1^{2}} - \frac{1}{n^{2}} \right\rbrack$
Where $n = 2,\ 3,\ 4,\ldots\ldots$
For the series limit of Lyman series $n = \infty$
$\therefore v_{1} = RC\left\lbrack \frac{1}{1^{2}} - \frac{1}{\infty^{2}} \right\rbrack = RC$ …(i)
For the first line of Lyman series, $n = 2$
$\therefore v_{2} = RC\left\lbrack \frac{1}{1^{2}} - \frac{1}{2^{2}} \right\rbrack = \frac{3}{4}RC$ …(ii)
For Balmer series $v = RC\left( \frac{1}{2^{2}} - \frac{1}{n^{2}} \right)$
Where $n = 3,\ 4,\ 5\ldots$
For the series limit of Balmer series $n = \infty$
$\therefore v_{3} = RC\left\lbrack \frac{1}{2^{2}} - \frac{1}{\infty^{2}} \right\rbrack = \frac{RC}{4}$ …(iii)
From equations (i), (ii) and (iii), we get
$v_{1} = v_{2} + v_{3} \Rightarrow v_{1} - v_{2} = v_{3}$