1. **State the problem:** A radioactive substance decays at a rate proportional to its mass. When the mass is 26 mg, the decay rate is 10 mg per week. We want to find a formula for the mass $M$ after $t$ weeks. 2. **Write the differential equation:** Since the decay rate is proportional to the mass, we have $$\frac{dM}{dt} = -kM$$ where $k > 0$ is the decay constant. 3. **Use the given data to find $k$:** When $M = 26$, the decay rate is 10 mg/week, so $$\frac{dM}{dt} = -10 = -k \times 26$$ 4. **Solve for $k$:** $$-10 = -26k \implies k = \frac{10}{26} = \frac{5}{13}$$ 5. **Solve the differential equation:** The general solution is $$M = M_0 e^{-kt}$$ where $M_0$ is the initial mass. 6. **Find $M_0$:** At $t=0$, $M = M_0$. Since the problem does not specify otherwise, assume $M_0 = 26$ mg. 7. **Write the final formula:** $$\boxed{M(t) = 26 e^{-\frac{5}{13} t}}$$ This formula gives the mass $M$ in mg after $t$ weeks.