The frequencies for series limit of Balmer and Paschen series are ' $v_1{ }^{\prime}$ and ' $v_3$ ' respectively. If frequency of first line of Balmer series ' $v_2$ ' then the relation between $v_1, v_2$ and $v_3$ is

The frequencies of spectral lines in hydrogen are given by the Rydberg formula: $v = R c \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$.

Series limit of Balmer series: $n_i = \infty$ to $n_f = 2$, so $v_1 = \frac{Rc}{4}$.

First line of Balmer series: $n_i = 3$ to $n_f = 2$, yielding $v_2 = \frac{5Rc}{36}$.

Series limit of Paschen series: $n_i = \infty$ to $n_f = 3$, giving $v_3 = \frac{Rc}{9}$.

The energy difference $E_\infty - E_2$ equals the sum $(E_\infty - E_3) + (E_3 - E_2)$. Dividing by Planck's constant $h$ gives $v_1 = v_3 + v_2$, or equivalently $v_1 - v_2 = v_3$.

Verifying numerically: $\frac{Rc}{4} - \frac{5Rc}{36} = \frac{9Rc - 5Rc}{36} = \frac{4Rc}{36} = \frac{Rc}{9}$, confirming $v_1 - v_2 = v_3$.

The correct relationship is thus $v_1 - v_2 = v_3$, corresponding to option C.