Doubly ionized lithium , $L{i}^2$, absorbs a photon and jumps from the ground state to its n=2level. What was the wavelength of the photon?

Whenever an electron absorbs energy enough for it to transition to higher state, it jumps to the higher state, and it loses energy while jumping down to the lower state from the higher state.

As you know that, energy of an electron in its nth orbit is given by

$E_n=-z^2\frac{13.6eV}{n^2}(n=1,2,3...........)$

Where, zis the atomic number of hydrogen-like atom and nis the principal quantum number.

Hence, if the ion jumps from ground state to n=2 , the energy will be,
$$\begin{gathered} E_2-E_1=3^2\frac{-13.6eV}{2^2}{3}^2-\frac{-13.6eV}{1^2} \\ =91.8eV \\ =1.47\times {10}^{-17}J \end{gathered}$$

Energy of the photon is given by,
$E=\frac{hc}{\lambda }.......\left(1\right)$

Here, h is the Planck’s constant, c is the speed of light, and $\lambda$is the wavelength.

The numerical value of hc is,

hc = 1240 eV .nm

Substitute known values into equation (1), and you have
$$\begin{gathered} 91.8eV=\frac{1240eV.nm}{\lambda } \\ \lambda =13.5nm \end{gathered}$$