= Solution
A stable terminal product obeys $\dot N_c=\lambda_bN_b$. Conservation of the number of chain members gives $N_a+N_b+N_c=N_{a0}+N_{c0}$ when $N_{b0}=0$. Therefore, for distinct <nuclear decay constants>,
$$
\boxed{N_c(t)=N_{c0}+N_{a0}\left[1-
\frac{\lambda_be^{-\lambda_at}-\lambda_ae^{-\lambda_bt}}
{\lambda_b-\lambda_a}\right].}
$$
Set $N_{c0}=0$ for an initially absent terminal product. For equal <nuclear decay constants>,
$$
\boxed{N_c(t)=N_{c0}+N_{a0}[1-(1+\lambda t)e^{-\lambda t}].}
$$
Both expressions have $\dot N_c(0)=0$ because the intermediate population initially vanishes; their leading growth is $\tfrac12N_{a0}\lambda_a\lambda_bt^2$. For positive <nuclear decay constants>, $N_c$ increases monotonically to $N_{c0}+N_{a0}$.
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