OG;
Yes, if one arbitrarily assumes ...
that all the mass captured in the "virtual scoop" uniformly gets a downwash velocity vv, the power required for this is Pi=0.5*mdot*vv^2. Equating this to the power Pi=Di*V to overcome the induced drag Di, ...
EDIT::
OTOH, if one equally arbitrarily assumes that the uniform downwash velocity is that predicted by Prandtl for the 'lifting line', the area corresponding to the lift momentum mdot*vv is twice as large.