I'm having some problems on how to model this situation correctly, using difference equations.
Say there's a medicine that has a half-life of 12 hours (every 12 hours, the amount of it on your blood is multiplied by 1/2) and 1g of this medicine is injected every 8 hours, also starting with 1g at $t=0$. Given this situation, I'm asked:
a) What will be the amount of medicine on a patient's blood after one day?
b) After 5 days?
My proposed model is: $y_{t}=\left(\frac{1}{2}\right)^{2/3}y_{t-1} + 1$, where $t$ jumps in eight hours steps. I can solve this recurrence equation but the answer I get is different from the answers my problem set gives me. I think I should solve and than evaluate it at $t=3$ for one day and $t=15$ for 5 days. Is my model wrong? Any help is welcome!
Thanks in advance!
I see nothing wrong with your model, nor with your general formula. Your answer for $t=3$ is also correct for after one day just after the 4th injection. (If it is just before the 4th injection then of course it would be $1$ gram less.)