= Solution
Let $A_{21}$ be the <spontaneous emission> coefficient, and let $B_{12}\rho$ and $B_{21}\rho$ be the absorption and <stimulated emission> rates per atom under the radiation spectral energy density $\rho$ at resonance. Because the two levels are nondegenerate, the <Einstein coefficients> satisfy $B_{12}=B_{21}=B$.
The <radiative balance in a nondegenerate two-level atom> is
$$
\frac{dN_2}{dt}=N_1B\rho-N_2B\rho-N_2A_{21}=0.
$$
The total spontaneous and stimulated emission rates are $R_{\rm sp}=N_2A_{21}$ and $R_{\rm st}=N_2B\rho$. Dividing the balance equation by $N_2B\rho$ gives
$$
\boxed{\frac{R_{\rm sp}}{R_{\rm st}}=\frac{A_{21}}{B\rho}
=\frac{N_1-N_2}{N_2}=\frac{N_1}{N_2}-1.}
$$
This presumes nonzero drive and upper-level population, and no other population-transfer mechanism. It does not require a thermal <Boltzmann distribution>. In this closed driven <two-level system>, spontaneous decay keeps $N_1>N_2$; arbitrarily intense driving approaches equal populations, rather than producing a <population inversion>.
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