Series and Limit Math Question

[Xboxdog2]

A doctor prescribes a 500-mg antibiotic tablet to be taken every eight hours. Just before each tablet is taken, 20% of the drug present in the preceding time step remains in the body.

(a) If Qn is the quantity of the antibiotic in the body just after the nth tablet is taken, write a difference equation that expresses Qn+1 in terms of Qn.

(b) Find a formula for Qn as a function of n.

(c) What quantity of the antibiotic remains in the body in the long run?

 

[MarkFL]

Hello, and welcome to FMH!

(a) I would write:

[MATH]Q_{n+1}=\frac{1}{5}Q_{n}+500[/MATH] where \(Q_1=500\)

This gives us the linear inhomogeneous difference equation:

[MATH]Q_{n+1}-\frac{1}{5}Q_{n}=500[/MATH]
So, what is the characteristic root/homogeneous solution, and what form will the particular solution take?

 

[ksdhart2]

Just in the interest of full disclosure, this exact problem was also posted on the AskMath subreddit, although this may be by another person (possibly another student in OP's class?), just judging from the wildly different usernames.

The hint provided here is actually a bit "nicer," in that it even explicitly answers part (a) for you, but I believe the other hint to actually be more useful because it encourages you to think about the problem and develop pattern recognition skills that will take you far in math.

 

[MarkFL]

To follow up (a little sooner than I normally would do because of hurricane Dorian):

We see the characteristic root is:

[MATH]r=\frac{1}{5}[/MATH]
And so the homogeneous solution is:

[MATH]h_n=c_1\left(\frac{1}{5}\right)^n=\frac{c_1}{5^n}[/MATH]
The particular solution will be of the form:

[MATH]p_n=A[/MATH]
Putting this into our difference equation, we obtain:

[MATH]A-\frac{1}{5}A=500\implies A=625[/MATH]
And so by the principle of superposition, we have the general solution:

[MATH]Q_n=h_n+p_n=\frac{c_1}{5^n}+625[/MATH]
To determine the value of the parameter, we may use:

[MATH]Q_1=\frac{c_1}{5}+625=500\implies c_1=-625[/MATH]
Hence:

[MATH]Q_n=-\frac{625}{5^n}+625=625\left(1-5^{-n}\right)[/MATH]
And from this, we can see:

[MATH]Q_{\infty}=\lim_{n\to\infty}Q_n=625[/MATH]