{Use of Tech} Periodic dosing
Many people take aspirin o...

Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.

c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

Accepted Answer

Hello, in this video, we are told that an individual is prescribed 50 mg of a drug every 8 hours. The drug's half-life is 8 hours, and we are going to let a N be the amount of drug present immediately after the nth dose. And if AN converges, what is its limit? So, if we take a look at what is given to us, we are told that we are working with 50 mg of a prescribed drug that is going to be taken every 8 hours. Now, the half-life of this drug is given to us as 8 hours. So what this means is that for whatever drug we're working with, which is going to be a minus 1. After 8 hours, we're only going to have half of that current dosage. Now, when we apply another dosage of the drug, since the drug has a total of 50 mg, that means that we're going to be adding 50 mg of the drug for every for every 8 hours. And so what this means is that we, if we come up with a recursive formula for every nth dosage, we can represent that as a N, which is going to equal to 1/2 AN minus 1 plus 50. This is going to be the equation that represents. The dosage that is going to be in our body every time we take a new dosage of the prescribed medicine. So, using this, how can we tell if this is going to be a convergent series, and what is its limit? Well, in order to figure out whether this is going to be a convergent series, we need to go ahead and take the limit as N approaches infinity. And so we're going to apply this limit to every term within this steady state. So we have the limit as N approaches infinity of a N is equal to the limit as N approaches infinity of 1/2 a N minus 1, plus the limit as N approaches infinity of 50. Now, by evaluating each of these terms, when we take the limit of the left hand side, that is just going to give us some random limit, which we will describe as L. And here, we will have the same characteristic on the first term on the right hand side, but we have this constant of 1/2. So we will have 1/2 L after evaluating the limit. And for the final term, we are taking the limit of a constant, which is just going to give us that constant. And so now what we want to go ahead and do is you want to solve for the limit, which is L. We will start by subtracting one half L to both sides of the equation, leaving us with one half L equal to 50, and by multiplying both sides of the equation by 2, we will have that the limit is equal to 100. And so what this means is that for the dosage, we do not want to exceed 100 mg. And that is how we approach this problem of finding the limit of this recursive formula, and the overall solution to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Geometric Sequences and Series

Limits of Sequences

Half-Life and Exponential Decay

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