The shortest wavelength of hydrogen atom in Lyman series is λ . The...

For Hydrogen atom ($H$) ($Z = 1$) : ${\left(\dfrac{1}{\lambda}\right)}_{H} = {R}_{H} \times {1}^{2}\left[\dfrac{1}{{1}^{2}} - \dfrac{1}{{\infty}^{2}}\right]$ ...(1)

For ${He}^{+}$ atom ($Z = 2$) : \left(\dfrac{1}{\lambda}\right){He}^{+} = {R}_{H} \times {2}^{2} \times $\left[\dfrac{1}{{2}^{2}} - \dfrac{1}{{3}^{2}}\right] = {R}_{H}\left[1 - \dfrac{4}{9}\right] = {R}_{H}\left(\dfrac{5}{9}\right)$ ..(2)

From (1) & (2)

$\dfrac{\lambda{He}^{+}}{{\lambda}_{H}} = \dfrac{9}{5}$

$\lambda{He}^{+} = \lambda \times \dfrac{9}{5}$

$\lambda{He}^{+} = \dfrac{9\lambda}{5}$.