1. Use Bohr transition energy relation

For a transition between two energy levels, the emitted photon energy is

$E = h\nu = \frac{hc}{\lambda}$

So, for the given transitions:

• From $A \to C$: $E_{AC} = \frac{hc}{2000\,\text{\AA}}$

• From $B \to C$: $E_{BC} = \frac{hc}{6000\,\text{\AA}}$

2. Relate the energy levels

Let the energy levels be $E_A, E_B, E_C$ with $E_A > E_B > E_C$.

Then,

$E_{AC} = E_A - E_C$ $E_{BC} = E_B - E_C$

The transition from $A \to B$ has energy

$E_{AB} = E_A - E_B$

But,

$E_A - E_B = (E_A - E_C) - (E_B - E_C)$

So,

$E_{AB} = E_{AC} - E_{BC}$

3. Substitute the photon energies

$\frac{hc}{\lambda_{AB}} = \frac{hc}{2000} - \frac{hc}{6000}$

Cancel $hc$:

$\frac{1}{\lambda_{AB}} = \frac{1}{2000} - \frac{1}{6000}$

Take LCM:

$\frac{1}{\lambda_{AB}} = \frac{3-1}{6000} = \frac{2}{6000} = \frac{1}{3000}$

Hence,

$\lambda_{AB} = 3000\,\text{\AA}$

4. Match with options

$\boxed{3000\,\text{\AA}}$

So the correct option is D.