The ratio of the wavelengths of the light absorbed by a Hydrogen atom when it undergoes \( n = 2 \rightarrow n = 3 \) and \( n = 4 \rightarrow n = 6 \) transitions, respectively, is

Using \( \frac{1}{\lambda} = R Z^2 (\frac{1}{n_1^2} - \frac{1}{n_2^2}) \), for \( 2 \rightarrow 3 \): \( \frac{1}{\lambda_1} = R(\frac{1}{4} - \frac{1}{9}) = \frac{5R}{36} \). For \( 4 \rightarrow 6 \): \( \frac{1}{\lambda_2} = R(\frac{1}{16} - \frac{1}{36}) = R(\frac{9-4}{144}) = \frac{5R}{144} \). Thus \( \frac{\lambda_1}{\lambda_2} = \frac{144}{36} = \frac{1}{4} \) (Wait, \( 36/144 = 1/4 \)).