The atomic hydrogen emits a line spectrum consisting of different series. The ratio of frequencies of the first line in Lyman and Balmer series is:

27 : 5

The correct answer is Option (3) → 27 : 5

Formula for frequency in hydrogen spectrum:

$\nu = R c \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

Lyman series: $n_1 = 1$, first line $\Rightarrow n_2 = 2$

$\nu_L = Rc \left( 1 - \frac{1}{4} \right) = Rc \cdot \frac{3}{4}$

Balmer series: $n_1 = 2$, first line $\Rightarrow n_2 = 3$

$\nu_B = Rc \left( \frac{1}{4} - \frac{1}{9} \right) = Rc \cdot \frac{5}{36}$

Ratio:

$\frac{\nu_L}{\nu_B} = \frac{\frac{3}{4}}{\frac{5}{36}} = \frac{3}{4} \cdot \frac{36}{5} = \frac{108}{20} = \frac{27}{5}$

Answer: The ratio of frequencies is $\frac{27}{5}$.