As per the given figure $$A$$, $$B$$ and $$C$$ are the first, second and third excited energy levels of hydrogen atom respectively. If the ratio of the two wavelengths (i.e. $$\frac{\lambda_1}{\lambda_2}$$) is $$\frac{7}{4n}$$, then the value of $$n$$ will be _____.

$$n_A = 2, \quad n_B = 3, \quad n_C = 4$$

$$\frac{1}{\lambda_1} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{9 - 4}{36} \right) = \frac{5R}{36}$$

$$\lambda_1 = \frac{36}{5R}$$

$$\frac{1}{\lambda_2} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{16 - 9}{144} \right) = \frac{7R}{144}$$

$$\lambda_2 = \frac{144}{7R}$$

$$\frac{\lambda_1}{\lambda_2} = \frac{\frac{36}{5R}}{\frac{144}{7R}}$$

$$\frac{\lambda_1}{\lambda_2} = \frac{36}{5} \times \frac{7}{144}$$

$$\frac{\lambda_1}{\lambda_2} = \frac{7}{5 \times 4} = \frac{7}{20}$$

$$\frac{7}{20} = \frac{7}{4n}$$

$$4n = 20$$

$$n = 5$$