Many people take a certain pain medication as a preventative measure for heart disease. Suppose a person takes 60 mg of the medication every 6 hr. Assume also that the medication has a half-life of 6 hr , that is, every 6 hr half of the drug in the blood is eliminated. Complete parts a. and b. below. a. Find a recurrence relation for the sequence \( \left\{d_{n}\right\} \) that gives the amount of drug in the blood after the nth dose, where \( d_{1}=60 \). A. \( d_{n+1}=2 d_{n}-60 \) B. \( d_{n+1}=2 d_{n}+120 \) C. \( d_{n+1}=\frac{1}{2} d_{n}-120 \) D. \( d_{n+1}=\frac{1}{2} d_{n}+60 \)

To create the recurrence relation for the amount of drug in the blood after the nth dose, we start with the initial dose. After the first dose (60 mg), after 6 hours, half of the drug is eliminated, which leaves 30 mg in the body. When the next dose of 60 mg is taken, we simply add that to what remains. So, the amount after the second dose would be the amount left plus the new dose: \( d_{2} = \frac{1}{2} \cdot 60 + 60 = 30 + 60 = 90 \). Following this logic, the recurrence relation can be expressed as: \[ d_{n+1} = \frac{1}{2} d_{n} + 60 \] Second Line: You might find this equivalently intriguing: the role of medication half-lives isn’t just a quirky number! Half-lives show how drugs are metabolized and eliminated from the body, so they have huge implications in medical contexts. For example, understanding the half-life helps doctors decide the right dosage and timing - essential for patient safety and effective treatment plans!