{Use of Tech} Drug Dosing
A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.




a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.
Find a recurrence relation for dₙ.

Verified step by step guidance

1. Understand the problem: The patient takes 75 mg of medication every 12 hours, and 60% of the medication is eliminated every 12 hours. We want to find a recurrence relation for the amount of medication in the bloodstream after n doses, denoted as \(d_n\), with \(d_1 = 75\) mg.
2. Identify what happens between doses: After each 12-hour period, 60% of the medication is eliminated, so 40% remains. This means the amount of medication just before taking the next dose is 40% of the previous amount, or \(0.4 \times d_{n-1}\).
3. Account for the new dose: At the time of the nth dose, the patient takes an additional 75 mg, which adds to the remaining medication from the previous dose.
4. Write the recurrence relation: Combining the remaining medication and the new dose, the amount after the nth dose is given by \[d_n = 0.4 \times d_{n-1} + 75\]
5. Confirm the initial condition: The first dose amount is given as \(d_1 = 75\), which fits the recurrence relation when \(n=1\).

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. In this problem, the amount of medication after each dose depends on the remaining amount from the previous dose plus the new dose. Understanding how to express this relationship mathematically is key to modeling the drug concentration over time.

Exponential Decay

Exponential decay describes how a quantity decreases by a consistent percentage over equal time intervals. Here, 60% of the medication is eliminated every 12 hours, meaning 40% remains. This decay factor is used to calculate the remaining drug amount before the next dose is added.

Initial Conditions in Sequences

Initial conditions specify the starting value of a sequence, which is essential for solving recurrence relations. Given d₁ = 75 mg, this sets the baseline amount of medication after the first dose, allowing subsequent terms to be computed accurately.