In Balmer series, wavelength of the \(2^{\text {nd }}\) line is ' \(\lambda_1\) ' and for Paschen series, wavelength of the \(1^{\text {st }}\) line is ' \(\lambda_2\) ', then the ratio ' \(\lambda_1\) ' to ' \(\lambda_2\) ' is

For spectral series, \(\frac{1}{\lambda}=R Z^2\left(\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right)\)
For the Balmer series, \(\mathrm{n}_1=2\)
The wavelength for \(2^{\text {nd }}\) line of the Balmer series is
\(
\begin{aligned}
& \frac{1}{\lambda_1}=\mathrm{RZ}^2\left(\frac{1}{2^2}-\frac{1}{4^2}\right) \\
& \frac{1}{\lambda_1}=\mathrm{RZ}^2\left(\frac{1}{4}-\frac{1}{16}\right) \\
& \frac{1}{\lambda_1}=\mathrm{RZ}^2\left(\frac{3}{16}\right) \Rightarrow \lambda_1=\left[\frac{16}{3}\right]
\end{aligned}
\)
For the Paschen series, \(\mathrm{n}_1=3\)
The wavelength for \(1^{\text {st }}\) line of the Paschen series is
\(\frac{1}{\lambda_2} =\mathrm{RZ}^2\left(\frac{1}{3^2}-\frac{1}{4^2}\right) \)
\( \frac{1}{\lambda_2} =\mathrm{RZ}^2\left(\frac{1}{9}-\frac{1}{16}\right) \)
\( \lambda_2 =\frac{144}{7} \)
\( \frac{1}{\lambda_2} =\mathrm{RZ}^2\left(\frac{7}{144}\right) \)
\( \therefore \frac{\lambda_1}{\lambda_2} =\frac{16}{3} \times \frac{7}{144}=\frac{7}{27}\)