Question

Periodic dosesSuppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

Accepted Answer

Identify the problem as a geometric series situation where doses are taken periodically and the drug decays exponentially between doses. The half-life of 6 hours means the drug amount halves every 6 hours, which matches the dosing interval. Express the amount of drug remaining from each dose at the time just before the next dose is taken. Since the half-life is 6 hours, the decay factor per 6 hours is $$r = \frac{1}{2}. $$Write the total steady-state amount of drug in the blood as the sum of an infinite geometric series where the first term $$a is $$the initial dose of 200 mg, and each subsequent term is multiplied by $$r = \frac{1}{2} to $$represent the remaining drug from previous doses. Set up the infinite series as $$S = 200 + 200 	imes \frac{1}{2} + 200 	imes \left(\frac{1}{2}\$$right)^2$$ + 200 	imes \left(\frac{1}{2}\$$right)^3$$ + \cdots$$ and recognize this as a geometric series with first term $$a = 200$$ and common ratio $$r = \frac{1}{2}. $$Use the formula for the sum of an infinite geometric series $$S = \frac{a}{1 - r}$$, where $$|r| \u003c 1$$, to express the steady-state amount of antibiotic in the blood.

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Key Concepts

Half-Life and Exponential Decay

Geometric Series and Infinite Sums

Steady-State Concentration in Pharmacokinetics