(i) Emission spectrum of Hydrogen atom: Lyman and Balmer series:

Lyman Series: When electrons will jump from higher energy orbit to $\mathit{n}=1$ orbit.
Balmer Series: When electrons will jump from higher energy orbit to $\mathit{n}=2$ orbit.
(ii) Longest wavelength of Lyman series:
$\frac{1}{{\mathit{\lambda }}_{\mathit{L}}}={\mathit{R}}_{\mathit{H}}\left[\frac{1}{1^2}-\frac{1}{2^2}\right]={\mathit{R}}_{\mathit{H}}\left[\frac{3}{4}\right]$
Shortest wavelength of Balmer series:
$\frac{1}{{\mathit{\lambda }}_{\mathit{B}}}={\mathit{R}}_{\mathit{H}}\left[\frac{1}{2^2}-\frac{1}{{\mathit{\infty }}^2}\right]={\mathit{R}}_{\mathit{H}}\left[\frac{1}{4}\right]$
Now, $\frac{{\mathit{\lambda }}_{\mathit{L}}}{{\mathit{\lambda }}_{\mathit{B}}}=\frac{\left[\frac{1}{4}\right]}{\left[\frac{3}{4}\right]}=\frac{1}{3}$