2025 RVHS P2 Q4

The dosage of medicine given to a patient needs to be carefully managed in order to achieve the desired result.

1. To give the patient time to adapt to a particular medicine, the dosage is slowly increased over time. The patient is initially given a dosage of \(0.75\) units. The dosage is increased by \(0.5\) units each day until a total of at least \(10\) units of medicine has been taken. Find the least number of days it takes to achieve this, and the dosage on that day.[4]
2. The doctor prescribed a new medication to another patient. After \(n\) days, the total dosage of the new medicine administered to the patient is given by \(\frac{3}{20} \left( \frac{5^n - 4^n}{5^{n-1}} \right)\) units.
   1. Show that the increase in dosage of the new medicine administered per day follows a geometric progression.[3]
   2. Find the total dosage of the new medicine administered to the patient after a long period of treatment.[2]
3. Doctors can manage the medication so that the daily dosage of medicine increases as an arithmetic progression. Explain in context why the arithmetic progression is not a preferred model for the dosage of medication.[1]