Preview Activity 8.2.1.
Warfarin is an anticoagulant that prevents blood clotting; often it is prescribed to stroke victims in order to help ensure blood flow. The level of warfarin has to reach a certain concentration in the blood in order to be effective. Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The drug is absorbed by the body and some is excreted from the system between doses. Assume that at the end of a 24 hour period, 8% of the drug remains in the body. Let \(Q(n)\) be the amount (in mg) of warfarin in the body before the \((n+1)\)st dose of the drug is administered.
(a)
Explain why \(Q(1) = 5 \times 0.08\) mg.
(b)
Explain why \(Q(2) = (5+Q(1)) \times 0.08\) mg. Then show that
\begin{equation*}
Q(2) = (5 \times 0.08)\left(1+0.08\right) \text{mg}\text{.}
\end{equation*}
(c)
Explain why \(Q(3) = (5+Q(2)) \times 0.08\) mg. Then show that
\begin{equation*}
Q(3) = (5 \times 0.08)\left(1+0.08+0.08^2\right) \text{mg}\text{.}
\end{equation*}
(d)
Explain why \(Q(4) = (5+Q(3)) \times 0.08\) mg. Then show that
\begin{equation*}
Q(4) = (5 \times 0.08)\left(1+0.08+0.08^2+0.08^3\right) \text{mg}\text{.}
\end{equation*}
(e)
There is a pattern that you should see emerging. Use this pattern to find a formula for \(Q(n)\text{,}\) where \(n\) is an arbitrary positive integer.
(f)
Complete the table below with values of \(Q(n)\) for the provided \(n\)-values (reporting \(Q(n)\) to 10 decimal places). What appears to be happening to the sequence \(Q(n)\) as \(n\) increases?
| \(n\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) |
| \(Q(n)\) | \(0.40\) |
