Electrons in a certain energy level n = n _ 1 , can emit 3 spectral...

Number of emitted spectral lines, $N = \dfrac{n ( n - 1 )}{2}$.

Case I
$N = 3$
$⇒ 3 = \dfrac{{n}_{1} ( {n}_{1} - 1 )}{2}$
$⇒ {n}_{1}^{2} - {n}_{1} - 6 = 0$
$⇒ ( {n}_{1} - 3 ) ( {n}_{1} + 2 ) = 0$
$⇒ {n}_{1} = 3$

Case II

$N = 6$
$⇒ 6 = \dfrac{{n}_{2} ( {n}_{2} - 1 )}{2}$
$⇒ {n2}^{2} - {n}_{2} - 12 = 0$
$⇒ ( {n}_{2} - 4 ) ( {n}_{2} + 3 ) = 0$
$⇒ {n}_{2} = 4$

Velocity of electron $v = \dfrac{{Ze}^{2}}{2{ε}_{0}hn}$
$∴ \dfrac{v1}{v2} = \dfrac{n2}{n1}$
$⇒ \dfrac{v1}{v2} = \dfrac{4}{3}$