When doctors prescribe medicine at equally spaced time intervals, they are aware that the body metabolizes the drug gradually. After some period of time, only a certain percent of the original amount remains. After each dose, the amount of the drug in the body is equal to the amount of the given dose plus the amount remaining from the previous doses. The amount of the drug present in the body after nth doses is modelled by a geometric series where t1 is the prescribed dosage and r is the previous dose remaining in the body.

Suppose a person with an ear infection takes a 200-mg ampicillin tablet every 4 h. About 12% of the drug in the body at the start of a four-hour period is still present at the end of that period. What amount of ampicillin is in the body, to the nearest tenth of a milligram,
(a) after taking third tablet?
(b) after taking the sixth tablet?

amount right now (1st pill) = 200 mg

amount after 2nd pill = 200 + .12(200) = 200(1.12)
amount after 3rd pill = 200 + 200(.12) + 200(.12)^2

amount after 4th pill = 200 + 200(.12) + 200(.12)62 + 200(.12)^3

...

amount after nth pill = 200 + 200(.12) + ... + 200(.12)^(n-1)

after 6th pill
amount = 200 + 200(.12) + ... + 200(.12)^5
= 200( 1 - .12^6)/(1-.12) = 227.27 mg

To solve this problem, we need to understand the concept of a geometric series and how it can be used to model the amount of a drug in the body after each dose.

In this scenario, the prescribed dosage of ampicillin is 200 mg, and each dose is taken every 4 hours. We are told that 12% of the drug remains in the body at the end of each 4-hour period.

Let's express this information in terms of a geometric series.

The first term (t1) of the series is the prescribed dosage, which is 200 mg.

The common ratio (r) is the percentage of the drug remaining after each 4-hour period. Since 12% remains, the common ratio (r) is 0.12.

Now, let's calculate the amount of ampicillin in the body after taking the third and sixth tablets:

(a) After taking the third tablet:
To find the amount of ampicillin in the body after the third tablet, we need to use the formula for the sum of a geometric series:

Sn = t1 * (1 - r^n) / (1 - r)

where Sn is the sum of the series after n terms.

Substituting the values, we have:
n = 3 (since we want to find the amount after the third tablet)
t1 = 200 mg
r = 0.12

Plugging these values into the formula:

S3 = 200 * (1 - 0.12^3) / (1 - 0.12)

Simplifying the expression gives us:
S3 ≈ 200 * (1 - 0.001728) / (0.88)
S3 ≈ 200 * (0.998272) / (0.88)
S3 ≈ 227.2 mg

Therefore, after taking the third tablet, there is approximately 227.2 mg of ampicillin in the body.

(b) After taking the sixth tablet:
Using the same formula, but with n = 6:

S6 = 200 * (1 - 0.12^6) / (1 - 0.12)

Calculating gives us:
S6 ≈ 200 * (1 - 0.000023) / (0.88)
S6 ≈ 200 * (0.999977) / (0.88)
S6 ≈ 249.9977 mg

Therefore, after taking the sixth tablet, there is approximately 250 mg of ampicillin in the body, to the nearest tenth of a milligram.

Keep in mind that these calculations assume a constant rate of metabolism and do not take into account factors such as absorption, elimination, or any other pharmacokinetic considerations. It's best to consult a healthcare professional for accurate and personalized information.