Drug Dosage Problem

A patient takes a 30-mg (milligram) antibiotic capsule every hour.At the end of any one hour, the amount of antibiotic remaining in her body is only 90% of the amount at the beginning of that hour. Your objective is to predict the total amount in her body after many hours.

The first 30 mg dose is taken at time t=0 h.How much of this dose remains at the end of 1

h? 2 h?3 h? 4 h?

Starting with the last dose, the amounts remaining from the first 3 doses (t=4h) can be written

30, 30(0.9), 30(0.9)², 30(0.9)³, 30(0.9)⁴

These numbers are part of a geometric sequence. The next term in the sequence is formed by multiplying the preceding term by 0.9, the common ratio.The total amount in the patient's body after these 5 doses is a partial sum of the geometric series,

30+ 30(0.9) + 30(0.9)²+30(0.9)³+30(0.9)⁴

How many milligrams of the antibiotic are in her body after these 5 doses?

You can use the SUM and SEQUENCE commands on your grapher to compute partial sums of any series.Write the appropriate commands to sum the formula 30(0.9)ⁿ from n=0 to n=t.Check your commands by showing that you get the same sum as in Problem 2 when t=4 h.

Find the 11^th partial sum (t=10) and the 21^st partial sum (t=20).

The partial sums for this series converge to a limit.What does this limit appear to be?What implications does the convergence of this series have for the patient in this problem?