In the hydrogen atom spectrum $${{\lambda _{3 - 1}}}$$  and $${{\lambda _{2 - 1}}}$$  represent wavelengths emitted due to transition from second and first excited states to the ground state respectively. The value of $$\frac{{{\lambda _{3 - 1}}}}{{{\lambda _{2 - 1}}}}$$  is

Solution :
$$\eqalign{ & \frac{1}{{{\lambda _{3 - 1}}}} = R\left( {\frac{1}{{{1^2}}} - \frac{1}{{{3^2}}}} \right) = \frac{{8R}}{9} \cr & \frac{1}{{{\lambda _{2 - 1}}}} = R\left( {\frac{1}{{{1^2}}} - \frac{1}{{{2^2}}}} \right) = \frac{{3R}}{4} \cr & \Rightarrow \frac{{{\lambda _{3 - 1}}}}{{{\lambda _{2 - 1}}}} = \frac{{27}}{{32}} \cr} $$