Explanation

(a) Number of emitted spectral lines
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N\ =$ $\frac{n(n - 1)}{2}$
Case I
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N = 3$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3 = \frac{n_{1}(n_{1} - 1)}{2}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n_{1}^{2}\ \ - \ n_{1}\ - 6\ = 0$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (n_{1}\ - 3)(n_{1}\ + 2) = 0$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n_{1}$=3
Case II
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N = 6$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6 = \frac{n_{2}(n_{2} - 1)}{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n_{2}^{2}$ –$n_{2} - 12\ = 0$
$\Rightarrow (n_{2} - 4)(n_{2}\ + 3) = 0$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n_{2} = 4\ ,\ n_{2}\ = - 3$
Again , as $n_{2}$ is always positive
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n_{2}$ = 4
Velocity of electron $v = \ \frac{{Ze}^{2}}{2\varepsilon_{0}hn}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{v_{1}}{v_{2}} = \frac{n_{2}}{n_{1}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{v_{1}}{v_{2}} = \frac{4}{3}$