Question

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes \(80 \mathrm{mg}\) of aspirin every 24 hours. Assume also that aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated. a. Find a recurrence relation for the sequence \(\left\\{d_{n}\right\\}\) that gives the amount of drug in the blood after the \(n\) th dose, where \(d_{1}=80\). b. Using a calculator, determine the limit of the sequence. In the long run, how much drug is in the person's blood? c. Confirm the result of part (b) by finding the limit of \(\left\\{d_{n}\right\\}\) directly.

Short answer

The long-run amount of aspirin in the person's blood is 160 mg.

Step-by-step solution

Setting up the recurrence relation

To set up the recurrence relation for the amount of drug in the blood after each dose, we first consider the given information: 1. The person takes a dose of \(80 \mathrm{mg}\) of aspirin after every \(24\) hours. 2. The half-life of aspirin is \(24\) hours. We are given that \(d_1 = 80\). Now, let's consider \(d_2\). The person takes a new \(80 \mathrm{mg}\) dose after \(24\) hours, so the remaining drug in the blood after \(24\) hours is half of the previous dose. Therefore, after the second dose, the amount of drug in the person's blood will be \(0.5d_1 + 80\). In general, when the person takes the \(n\)th dose, the amount of drug in the person's blood would be half of what was left from the previous dose plus the new \(80 \mathrm{mg}\) dose. So we can write the recurrence relation as: \(d_{n} = 0.5d_{n-1} + 80\) for \(n \geq 2\). Now let's find the limit of this sequence.

Finding the limit using a calculator

To find the limit of this sequence, we can use a calculator to compute the first few terms of the sequence and see if it converges to a particular value. \(d_1 = 80\); \(d_2 = 0.5d_1 + 80 = 0.5(80) + 80 = 120\); \(d_3 = 0.5d_2 + 80 = 0.5(120) + 80 = 140\); \(d_4 = 0.5d_3 + 80 = 0.5(140) + 80 = 150\); \(d_5 = 0.5d_4 + 80 = 0.5(150) + 80 = 155\); ... After several iterations, we can see that the sequence seems to converge to \(160 \mathrm{mg}\). This means that in the long run, the person will have \(160 \mathrm{mg}\) of aspirin in their blood. Now, let's confirm this result by finding the limit directly.

Confirming the limit directly

To find the limit directly, we can set up the following equation $$ \lim_{n \to \infty} d_n = L $$ Since the limit exists, we know that $$ \lim_{n \to \infty} d_{n-1} = L $$ From the recurrence relation, we have $$ d_n = 0.5d_{n-1} + 80 $$ Taking the limit on both sides $$ \lim_{n \to \infty} d_n = 0.5\lim_{n \to \infty} d_{n-1} + 80 $$ Substituting the limits (\(L\)) we get $$ L = 0.5L + 80 $$ Solving for \(L\), $$ L - 0.5L = 80 \\ 0.5L = 80 \\ L = 160 $$ So, the limit, and thus the long-run amount of drug in the person's blood, is indeed \(160 \mathrm{mg}\), confirming our earlier result.

Key concepts

Aspirin Dosage

When a person takes aspirin regularly, the body processes it at a certain rate. In this scenario, an individual takes an 80 mg aspirin dose every 24 hours. This creates a sequence where the amount of aspirin in the bloodstream changes over time. Essentially, how much aspirin remains in the blood after each dose depends on both the new dose and the amount remaining from previous doses.

Understanding this process is crucial for determining a safe and effective regimen. Too much aspirin in the body could be harmful, while too little might not provide the intended benefits. Thus, the determination of safe dosage intervals is based largely on the drug's half-life.

For calculations:

• The amount of aspirin initially added is always 80 mg after each dose.
• The remaining dose decreases due to the half-life effect.

Half-life

The half-life refers to the time it takes for the concentration of a substance in the body to reduce to half its initial amount. In the context of aspirin, the half-life is 24 hours. This means after 24 hours, only half of the aspirin remains in the bloodstream.

This characteristic plays a critical role in designing dosage schedules. It ensures that the remaining amount of aspirin does not accumulate to toxic levels. In our exercise, after each dose, the amount of aspirin decreases by half of what was present before the new dose is taken.

• For instance, if one initially has 80 mg, the next day halves it to 40 mg, before adding the new dose.
• This pattern persists, guiding the creation of the recurrence relation that models aspirin concentration over time.

Limit of a Sequence

In mathematics, the limit of a sequence is what the sequence tends to as the number of terms increases indefinitely. In our task, we're interested in the sequence of aspirin levels in the blood over days of continuous dosing, as represented by the recurrence relation: \[d_{n} = 0.5d_{n-1} + 80\]Finding the limit of this sequence helps us understand the long-term concentration of aspirin in the bloodstream. By calculating a few terms, we notice that the sequence approaches 160 mg, indicating that this is the limit.

Calculating the limit involves setting up and solving the equation: \[L = 0.5L + 80\]By solving, we find that the sequence converges to a limit (\(L\)) of 160 mg. This value suggests that, over time, the patient's bloodstream stabilizes at 160 mg of aspirin provided the regimen is maintained. This equilibrium represents a balance between the dose taken and the half-life effect.