Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths $$r = \frac{\lambda_{1}}{\lambda_{2}}$$, is given by:

$$\Delta E = \frac{hc}{\lambda} \implies \lambda \propto \frac{1}{\Delta E}$$

$$\frac{\lambda_1}{\lambda_2} = \frac{\Delta E_2}{\Delta E_1}$$

For transition $$\lambda_2$$: The transition occurs from level $$-E$$ to level $$-\frac{4}{3}E$$.

$$\Delta E_2 = | -E - (-\frac{4}{3}E) | = | -E + \frac{4}{3}E | = \frac{1}{3}E$$

$$\Delta E_1 = | -E - (-2E) | = | -E + 2E | = E$$

$$r = \frac{\lambda_1}{\lambda_2} = \frac{\Delta E_2}{\Delta E_1}$$

$$r = \frac{\frac{1}{3}E}{E}$$

$$r = \frac{1}{3}$$