Given:  $E_1$ is the energy for the transition from $n=2\to n=1$ and $E_2$ is the energy for the transition from $n=3\to n=2$.

Now the excitation energy is given as, 

$∆E=-13.6 \left(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2}\right) ev$

For the first case, $E_1\alpha \left(\frac{1}{1^2}-\frac{1}{2^2}\right)=\frac{3}{4}$

For the second case, $E_2\alpha \left(\frac{1}{2^2}-\frac{1}{3^2}\right)=\frac{5}{36}$

 Thus the ratio is,  $\frac{E_2}{E_1}\alpha \frac{5}{27}$.