In the spectrum of the hydrogen atom, the ratio of the wavelengths...

When an electron jumps& down from any higher energy level to the first energy level, then the emitted lines form the Lyman series

$\dfrac{1}{\lambdaL} = R \left(\dfrac{1}{12} - \dfrac{1}{n2}\right), where$Ris the Rydberg constant.</p><p>Similarly, when an electron~jumps down& from any higher energy level to the second energy level, then the emitted lines form the Balmer series</p><p>\dfrac{1}{\lambdaB} =R \left(\dfrac{1}{22} - \dfrac{1}{n2}\right)$<br /><br />For maximum wavelength, the energy gap should be the smallest</p><p>$n = 2$,~$\dfrac{1}{\lambdaL} = R\left(1 - \dfrac{1}{22}\right) = R\left(1 - \dfrac{1}{4}\right) = \dfrac{3R}{4}$</p><p>$n = 3$,~$\dfrac{1}{\lambdaB} = R\left(\dfrac{1}{22} - \dfrac{1}{32}\right) = \dfrac{5R}{36}$</p><p>$\dfrac{{λ}_{L}}{\lambdaB} = \dfrac{5}{27}$