{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.




c.Find the limit of the sequence. What is the physical meaning of this limit?

Verified step by step guidance

1. Identify the sequence that models the amount of medication in the blood just before each dose. Let \( a_n \) represent the amount of medication in the blood just before the \( n^{th} \) dose is taken.
2. Recognize that after each 12-hour period, 60% of the medication is eliminated, so 40% remains. This means the amount of medication remaining after elimination is \( 0.4 \times a_n \). Then, the patient takes an additional 75 mg, so the next term in the sequence is \( a_{n+1} = 0.4 \times a_n + 75 \).
3. Understand that this is a linear recurrence relation of the form \( a_{n+1} = r a_n + d \), where \( r = 0.4 \) and \( d = 75 \). To find the limit of the sequence as \( n \to \infty \), we look for the steady-state value \( L \) where \( a_{n+1} = a_n = L \).
4. Set up the equation for the limit: \( L = 0.4 L + 75 \). Solve this equation for \( L \) to find the long-term amount of medication in the blood just before taking a dose.
5. Interpret the physical meaning of the limit \( L \): it represents the equilibrium concentration of the medication in the blood, where the amount eliminated and the amount taken balance out, resulting in a stable level of medication over time.

Key Concepts

Geometric Sequences and Series

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. Understanding how to find the limit of such sequences is essential when analyzing repeated processes like drug dosing, where the amount changes by a fixed percentage over time.

Limit of a Sequence

The limit of a sequence describes the value that the terms of the sequence approach as the number of terms goes to infinity. In drug dosing, this limit represents the steady-state concentration of the medication in the blood after many doses.

Pharmacokinetics and Drug Elimination

Pharmacokinetics studies how drugs are absorbed, distributed, metabolized, and eliminated by the body. The elimination rate (60% every 12 hours) affects the drug concentration over time, influencing the calculation of the steady-state level in repeated dosing.