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. \( d_{n+1}=\frac{1}{2} d_{n}+60 \) b. Using a calculator, determine the limit of the sequence. In the long run, how much drug is in the person's blood? Confirm the result by finding the limit of the sequence directly. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The limit of the sequence is B. The limit does not exist.

To tackle the recurrence relation, let’s think about the bloodstream levels after each dose. Each dose adds a certain amount of drug while also considering the rate at which the drug is metabolized or eliminated from the body. If we assume that 60 units of the drug are metabolized after each dose, we can derive that \( d_{n+1} = 2d_n - 60 \) accurately reflects how the drug accumulates (the 60 reduced from the amount present from the previous dose). As for the long-term limit of the sequence, we’ll set \( L \) as the limit of \( d_n \) as \( n \) approaches infinity. Utilizing the recurrence relation and solving \( L = 2L - 60 \) leads us to the conclusion that \( L = 60 \). So, in the long run, the amount of drug in the person's blood stabilizes at 60 units. The limit of the sequence is indeed 60.