Hydrogen Spectrum & Spectral Series question from MHT CET 2023 (10 May, Shift 1)

Answer: (a)

1. Paschen series: $n_1 = 3$.
2. Longest ($4 \to 3$): $\frac{1}{\lambda_{max}} = R\left(\frac{1}{9}-\frac{1}{16}\right) = \frac{7R}{144}$
3. Shortest ($\infty \to 3$): $\frac{1}{\lambda_{min}} = \frac{R}{9}$
4. $\frac{\lambda_{max}}{\lambda_{min}} = \frac{144/7}{9} = \frac{144}{63}$
5. So the answer is (a) $\frac{144}{63}$.