In hydrogen spectrum, the shortest wavelength in the Balmer series is $\lambda$. The shortest wavelength in the Bracket series is

Shortest wavelength is the maximum energy, i.e., transition from infinity to 2^nd (Balmer series)
$\frac{\mathrm{hc}}{\lambda }=13.6\left(\frac{1}{4}-\frac{1}{\mathrm{\infty }}\right)=\frac{13.6}{4}$
(For Brackett series, n=4)
$\frac{\mathrm{hc}}{{\lambda }^1}=13.6\left(\frac{1}{16}-\frac{1}{\mathrm{\infty }}\right)=\frac{13.6}{16}$
dividing the two ${\lambda }^1=4\lambda$