a. Find a recurrence relation for the sequence \( \left\{d_{n}\right\} \) that gives the amount of drug in the blood after the \( n \)th dose, where \( d_{1}=80 \). A. \( d_{n+1}=2 d_{n}+160 \) B. \( d_{n+1}=\frac{1}{2} d_{n}+80 \) C. \( d_{n+1}=2 d_{n}-80 \) D. \( d_{n+1}=\frac{1}{2} d_{n}-160 \)

To find the recurrence relation for the sequence \( \left\{d_{n}\right\} \) that gives the amount of drug in the blood after the \( n \)th dose where \( d_{1}=80 \), we need to consider how the drug behaves in the bloodstream after each dose. After the first dose, we have \( d_1 = 80 \). If we assume that the drug decays or gets metabolized over time and a new dose of 80 is added with each subsequent dose, we can establish that each subsequent amount is influenced by half of the previous amount plus the new 80 mg dose. Thus, the most appropriate recurrence is: B. \( d_{n+1}=\frac{1}{2} d_{n}+80 \). This denotes that after each new dose, the previous amount in the system is halved, and a new dose is added. Involving drug dosages can lead to some fascinating discussions! The concept of drug accumulation and half-life in pharmacokinetics introduces you to the fascinating interplay of biology and mathematics. As a fun fact, different drugs have varying half-lives, which affects how quickly or slowly they act in the body, influencing everything from medicine dosing schedules to patient recovery times. If you’re keen to delve deeper into this topic, consider exploring resources about pharmacology and drug interactions. Books and articles can provide a richer perspective on how drug dosages are determined, including individual patient factors such as metabolism, age, and body weight—all of which can drastically affect the outcomes and efficacy of treatments!