For the stortest wavelength $\lambda$; of Balmer series, $\mathrm{n}_1=$
2 and $n_2=\infty$
$\begin{aligned}
& \frac{1}{\lambda_1}=\left(\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right) . \\
& \frac{1}{\lambda_1}=\frac{\mathrm{R}}{4} ...(1)
\end{aligned}$
For the longest wavelength $\lambda_2$ of Balmer series
$\frac{1}{\lambda_2}=\mathrm{R}\left(\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right) \quad \mathrm{n}_1=2 \text { and } \mathrm{n}_2=3$
$\begin{aligned} & \frac{1}{\lambda_2}=\mathrm{R}\left(\frac{1}{2^2}-\frac{1}{3^2}\right) \\ & \frac{1}{\lambda_2}=\mathrm{R}\left(\frac{1}{4}-\frac{1}{9}\right) \\ & \frac{1}{\lambda_2}=\mathrm{R}\left(\frac{9-4}{36}\right)\end{aligned}$
$\frac{1}{\lambda_2}=\frac{5 \mathrm{R}}{36}$ ...(2)
Subtracting equation (2) by (1), we have
$\begin{aligned}
& \frac{1}{\lambda_1}-\frac{1}{\lambda_2}=\frac{\mathrm{R}}{4}-\frac{5 \mathrm{R}}{36}=\frac{9 \mathrm{R}-5 \mathrm{R}}{36}=\frac{4 \mathrm{R}}{36} \\
& =\frac{\mathrm{R}}{9}
\end{aligned}$
Rydberg constant, $\mathrm{R}=\frac{9}{\lambda_1}-\frac{9}{\lambda_2}$