Chapter 7.5, Problem 47PS

Solution

a.

To state: A recursive rule for the amount of the drug in the bloodstream after n doses if a person repeatedly takes 20 milligrams of a prescribed drug every 4 hours.

The resultant rule is $a_1=20,a_n=0.7a_{n-1}+20$ .

Given information:

A person repeatedly takes 20 milligrams of a prescribed drug every 4 hours. Thirty percent of the drug is removed the bloodstream every 4 hours.

Explanation:

Consider the given information:

A person repeatedly takes 20 milligrams of a prescribed drug every 4 hours. It means $a_1=20$ .

Thirty percent of the drug is removed the bloodstream every 4 hours. That means only 70 percent drug is left from the previous dose.

Now form a recursive rule $a_1=20,a_n=0.7a_{n-1}+20$:

Therefore, the recursive rule is $a_1=20,a_n=0.7a_{n-1}+20$ .

b.

To state: The maintenance level of this drug, given a dosage of 20 milligrams if the value that a drug level in a person's body approaches after an extended period of time is called the maintenance level.

The resultant maintenance level is $\frac{200}{3}$ milligrams.

Given information:

A drug level in a person's body approaches after an extended period of time is called the maintenance level.

Explanation:

To find the maintenance level, find d such that $a_n+d$ becomes a geometric series,

  $\begin{array}{c}{a}_n+d=0.7a_{n-1}+20+d\\ =0.7\left(a_{n-1}+\frac{20+d}{0.7}\right)\\ d=\frac{20+d}{0.7}\\ d=-\frac{200}{3}\end{array}$

  $a_n-\frac{200}{3}=0.7\left(a_{n-1}-\frac{200}{3}\right)$

The series will converge to $\frac{200}{3}$ .

Therefore, the maintenance level of this drug, given a dosage of 20 milligrams is $\frac{200}{3}$ milligrams.

c.

To state: The affect of doubling the dosage on the maintenance level.

The resultant maintenance level is $\frac{400}{3}$ milligrams.

Given information:

Given dosage is 20 milligrams. Doubling the dosage of the drug will result in 40 milligrams.

Explanation:

To find the dosage affect on maintenance level, find d such that $a_n+d$ becomes a geometric series with doubled dosage

  $\begin{array}{c}{a}_n+d=0.7a_{n-1}+40+d\\ =0.7\left(a_{n-1}+\frac{40+d}{0.7}\right)\\ d=\frac{40+d}{0.7}\\ d=-\frac{400}{3}\end{array}$

  $\begin{array}{c}{a}_n-\frac{400}{3}=0.7\left(a_{n-1}-\frac{400}{3}\right)\\ a_n\to \frac{400}{3}\end{array}$

The series will converge to $\frac{400}{3}$ .

Therefore, the maintenance level of this drug, given a dosage of 40 milligrams is $\frac{400}{3}$ milligrams.