In Paschen series, wavelength of first line is ' $\lambda_1$ ' and for Brackett series, wavelength of first line is ' $\lambda_2$ ' then ratio $\frac{\lambda_1}{\lambda_2}$ is

The Rydberg formula gives the wavelength of spectral lines as

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

For the first line of the Paschen series ($n_f = 3$, $n_i = 4$):

$\frac{1}{\lambda_1} = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{7}{144} \right)$

Hence $\lambda_1 = \frac{144}{7R}$.

For the first line of the Brackett series ($n_f = 4$, $n_i = 5$):

$\frac{1}{\lambda_2} = R \left( \frac{1}{16} - \frac{1}{25} \right) = R \left( \frac{9}{400} \right)$

Thus $\lambda_2 = \frac{400}{9R}$.

The ratio becomes

$\frac{\lambda_1}{\lambda_2} = \frac{144}{7R} \times \frac{9R}{400} = \frac{1296}{2800}$

Simplifying by dividing numerator and denominator by 16 yields $\frac{81}{175}$.

This corresponds to option C.