Module 7, Recursively-Defined Sequences & Iterations Assignment You have learned how to write both the explicit and recursive forms of arithmetic and geometric sequences. You can solve for a term directly using the explicit form. In a recursive form, you need to know the value of all the terms that come before the term you are looking for. For example, to find the value of the 10th term in a recursively-defined sequence, you need to know the values of term 1 to term 9. Finding the values of terms in recursively-defined sequences is similar to the process of iteration. Both recursively-defined sequences and iteration use their own notation.

Consider the scenario below.

This morning Haidar noticed that her dog, Aldi, seemed unwell. She took him to the vet, where he was diagnosed with an ear infection. The veterinarian prescribed an antibiotic called amoxicillin to help Aldi feel better. Haidar will give Aldi 150 mg of the antibiotic once each day for 10 days. Each time Haidar gives Aldi his next dose of medication, 40% of the previous dose remains in his bloodstream.

Use what you know about recursive sequences and iteration to explore the mathematics related to this scenario.

1. Determine the amount of medication that is in Aldi’s bloodstream after Haidar gives him each of the 1st, 2nd, 3rd, and 4th doses of amoxicillin. Show your work.

2. Use your response to question 1 to write a recursive definition for the amount of medication in Aldi’s bloodstream after each dose of medication. Is your definition arithmetic, geometric, or neither? Explain.

3. Convert your recursive definition into iteration notation. Explain how you made your conversion.

4. Use iterative form to give its first 3 iterates. Explain how you determined your answer.

5. Use your iterative form to find out how much medication is in Aldi’s bloodstream right before Haidar gives him his last dose of medication. Show your work.

6. Examine your answer to question 5. Does the amount of medication in Aldi’s system seem to be approaching a certain value? If so, what is the value? Use Desmos to plot the points that represent the first 9 iterates. Provide a screenshot of your plot. What kind of shape do you see in the points?

7. The actual half-life of amoxicillin is about 62 minutes. Recall that half-life is the amount of time it takes for a substance to decay to half of its original amount. How long after Haidar gives Aldi his last dose of medication will it take before the amount of medication in Aldi’s bloodstream is effectively 0? Describe how you determined your solving method.