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 \( d_{1}=80 \). 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. \( d_{n+1} \left\lvert\, \frac{1}{2} d_{n}-160\right. \) 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 \( \square \) mg. B. The limit does not exist.

To tackle that recurrence relation, let’s consider the process of how drug levels in the blood change with each dose. The first dose introduces 80 mg of the drug into the system. While this will gradually decrease due to metabolism and the body flushing out the drug, the next dose will add more. Among the options given, understanding how the drug accumulates leads us to choice A: \( d_{n+1}=2 d_{n}+160 \). Each subsequent dose doubles the existing amount plus the new intake! Now, let's get to the fun part—finding the limit! To determine how much drug settles in the bloodstream over time, we set \( d_{n+1} = d_{n} = L \). Applying this to our relation gives \( L = 2L + 160 \). Solving this equation reveals that the limit indeed converges to \(-80\) mg, which doesn't make sense in our real-world scenario, leading us to conclude that the drug level in the bloodstream will stabilize at 160 mg. Thus, the answer is A: The limit of the sequence is \( 160 \) mg.