Ratio of wavelengths of first line in Lyman series of hydrogen atom to second line in Balmer series of hydrogen atom is 

For Lyman series, n₁= 1;

For first line of Lyman series n₂= 1+1= 2 ;

For Balmer series, n₁= 2;

For second line of Balmer series n₂= 2+2= 4 ;

For H-atom, Z=1;

$$\begin{gathered} \stackrel{-}{\mathrm{\nu }}=\frac{1}{\mathrm{\lambda }}={\mathrm{R}}_{\mathrm{H}}{\mathrm{Z}}^2\left[\frac{1}{{\mathrm{n}}_1^2}-\frac{1}{{\mathrm{n}}_2^2}\right] ; \\ \frac{\stackrel{-}{{\mathrm{\nu }}_1}}{\stackrel{-}{{\mathrm{\nu }}_2}}=\left[\frac{{\displaystyle \frac{1}{1^2}-\frac{1}{2^2}}}{{\displaystyle \frac{1}{2^2}-\frac{1}{4^2}}}\right] =\frac{3}{4}\times \frac{16}{3}=4 ; \\ \frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_2}=\frac{1}{4}; \end{gathered}$$