Nuclei of a radioactive element $$A$$ are being produced at a constant rate $$\alpha$$. The element $$A$$ has a decay constant $$\lambda$$. If there are $$N_0$$ nuclei at $$t = 0$$, then

number of nuclei $$N(t)$$, at time $$t$$, is $$N(t) = \frac{1}{\lambda}\left[\left(\alpha - \lambda N_0\right)e^{-\lambda t}\right]$$

if $$\alpha = \lambda N_0$$, the number of nuclei $$N(t)$$ at any time $$t$$ will remain constant

if $$\alpha = 2\lambda N_0$$ then $$N(t) = 2N_0$$ as $$t \rightarrow \infty$$

if $$\alpha = 2\lambda N_0$$, the number of nuclei $$N(t)$$ after one half-life of $$A$$ is $$N\left(\frac{T}{2}\right) = \frac{3}{2}N_0$$