The ratio of the frequency corresponding to the third line in Lyman series of hydrogen atomic spectrum to that of the first line in Balmer series of Li²⁺ spectrum is

Lyman series, n₁ = 1
For third line of Lyman series, n₂ = 4
For hydrogen, Z= 1

$\begin{array}{l}{v}_H=\frac{1}{\lambda }=R_H{Z}^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\\ =R_H(1{)}^2\left(\frac{1}{1}-\frac{1}{(4{)}^2}\right)=\frac{15}{16}{R}_H .... (i)\end{array}$

For lithium, Z=3
For first line of Balmer series, n₁ = 2,n₂ = 3

$\begin{array}{l}{v}_{\mathrm{Li}}=R_{\mathrm{H}}(3{)}^2\left(\frac{1}{(2{)}^2}-\frac{1}{(3{)}^2}\right)=R_{\mathrm{H}}\times 9\times \frac{5}{56}\\ =\frac{5}{4}{R}_{\mathrm{H}} . . . .\left(ii\right)\end{array}$

on dividing equation (i) by (ii), we get $\frac{v_{\mathrm{H}}}{v_{\mathrm{Li}}}=\frac{(15/16){\mathrm{R}}_{\mathrm{H}}}{(5/4){\mathrm{R}}_{\mathrm{H}}}=\frac{15}{16}\times \frac{4}{5}=\frac{3}{4}$