Ratio of longest wavelengths corresponding to Lyman and Balmer...

The transition equation for Lyman series is given by $\frac{1}{\lambda}=R\left(\frac{1}{1^2}-\frac{1}{n^2}\right)$ for largest wavelength, $n=2$. $\therefore \quad \frac{1}{\lambda_{\max }}=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)$ The transition equation for Balmer series is given by $\frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{n^2}\right)$ for largest wavelength, $n = 3$ $\therefore \quad \frac{1}{\lambda_{\max }}=R\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$ Therefore, $\frac{\lambda_{L_{\max }}}{\lambda_{B_{\max }}}=\frac{\left(\frac{1}{2^2}-\frac{1}{3^2}\right)}{\left(\frac{1}{1^2}-\frac{1}{2^2}\right)}=\frac{\frac{5}{36}}{\frac{3}{4}}=\frac{5}{27}$