What is the ratio of the frequencies of the long wavelength limits of…

The correct answer is Option (1) → 5 : 27

According to Rydberg formula,

$\frac{1}{λ}=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$

for longest $λ$ of lyman series, $(n_f=1,n_i=2)$

$\frac{1}{λ_1}=R_H\left(\frac{1}{1^2}-\frac{1}{4}\right)$

$⇒λ_1=\frac{4}{3R_H}$

for longest $λ$ of Balmer series, $(n_f=2,n_i=3)$

$⇒\frac{1}{λ_2}=R_H\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$

$⇒λ_2=\frac{36}{5R_H}$

$∴\frac{λ_1}{λ_2}=\frac{4}{3R_H}×\frac{5R_H}{36}=\frac{5}{27}$