72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence


c.Find a recurrence relation that generates the sequence.


Drug elimination
Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

Verified step by step guidance

1. Identify the initial condition given: the amount of drug at time zero is \(d_0 = 200\) mg.
2. Understand the process: every hour, 5% of the drug is eliminated, meaning 95% remains after each hour.
3. Express the amount of drug remaining after one hour in terms of the previous amount: \(d_1 = 0.95 \times d_0\).
4. Generalize this relationship to a recurrence relation for any hour \(n\): \(d_n = 0.95 \times d_{n-1}\) for \(n \geq 1\).
5. Summarize the recurrence relation with the initial condition: \(d_0 = 200\) and \(d_n = 0.95 \times d_{n-1}\).

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using one or more previous terms. It provides a way to model processes that evolve step-by-step, such as drug concentration over time. Understanding how to express the current amount based on the previous amount is key to formulating the relation.

Exponential Decay

Exponential decay describes a quantity that decreases by a fixed percentage over equal time intervals. In this problem, 5% of the drug is eliminated each hour, meaning 95% remains. Recognizing this helps to model the drug amount as a sequence that decreases multiplicatively.

Initial Conditions in Sequences

Initial conditions specify the starting value of a sequence, which is essential for solving recurrence relations. Here, d₀ = 200 mg sets the initial drug amount in the bloodstream, anchoring the sequence and allowing calculation of subsequent terms.