The longest wave length of hydrogen atom in Lyman series is $\lambda$. The shortest wavelength in Balmer series of He⁺ is

For H:-  $\frac{1}{{\lambda }_H}=R_H[\frac{1}{1^2}-\frac{1}{2^2}]\times 1^2$
For  $H{e}^{+}:\frac{1}{{\lambda }_{H{e}^{+}}}=R_H[\frac{1}{2^2}-\frac{1}{{\alpha }^2}]\times 2^2$
$\frac{1}{{\lambda }_H}=\frac{3R_H}{4}$  …… (1)
$\frac{1}{{\lambda }_{H{e}^{+}}}=R_H\left[\frac{1}{4}\right]\times 4=R_H$  ….. (2)
From (1) & (2) $\frac{\lambda H{e}^{+}}{\lambda }=\frac{1}{R_H}.\frac{3R_H}{4}$
$\lambda H{e}^{+}=\frac{3\lambda }{4}$