I'm not sure if I get this right, some pointers could be helpful.
Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3.75mg will remain in the blood. But by then you'd have to take another dose, increasing the blood level to 63.75mg.
Another 24 hours passes and you'd have 3.984375 mg in your blood. This is again increased by 60mg et cetera.
So this is modeled by:
blood_level = (60 + level_previous_day) * 0.5^4
Am I right?
This seems to tell me the level in the blood will never excede 4 milligrams by the end of any given day. In other words: the limit of this recurrence relation will be 4.
Are these assumptions correct? Am having a hard time matching my intuition with this result to be honest...
Your recurrence is perfect. So Blood level at $n^{th}$ day is given by :- $$T(n)=(60+T(n-1))\left(\frac{1}{2}\right)^4 \tag{1.}$$ $$T(0)=0$$ From equation 1 :- $$T(n)=60\left(\frac{1}{2}\right)^{4\times1}+60\left(\frac{1}{2}\right)^{4\times2}+...........+60\left(\frac{1}{2}\right)^{4\times k}+T(n-k)\left(\frac{1}{2}\right)^{4\times k}\tag{2.}$$ putting $k=n$ in eq. 2 :- $$T(n)=60\left[\frac{\frac{1}{2^4}\left(\frac{1}{2^{4n}}-1\right)}{\frac{1}{2^4}-1}\right]=4\left(1-\frac{1}{2^{4n}}\right)$$$$\tag{sum of n terms of GP}$$ If eq 2 be infinite series (G.P.) then it would be $T(n)=4$