Greenwood press

116
SEQUENCES

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SERIES
117
SERIES
Many applications that are based on the sum of predictable discrete patterns
can be examined with series. For example, a doctor may prescribe an amount of
medication to take each day, because he or she knows that the patient’s blood-
stream will be able to maintain a certain level of the medication over time.
Prescriptions are based on a mathematical series, because the total amount of
drug accumulates in the bloodstream each day. In other words, the sum of the
remaining amounts of the drug in the bloodstream is added to a new amount
everyday. One way to determine the total amount of a drug that will eventually
end up in the bloodstream is to take the initial amount and add the amount that
remains from yesterday, from two days ago, three days ago, and so on. If the
amount of drug that remains in the bloodstream is a predictable pattern each day,
then an equation can be used to compare dosages and accumulating amounts in
the bloodstream.
Some illnesses, such as high blood pressure or thyroid deficiency, can be
treated with regular medication. Suppose a doctor knows that 200 mg of a drug
is the amount of medication needed to maintain the patient’s health. Because
most drugs circulate in the bloodstream, amounts of the drug are removed as the
blood is cleaned by the kidneys. Suppose that the kidneys remove 40 percent of
the drug each day. That leaves the drug effectiveness at 60 percent of what it was
twenty-four hours earlier. Therefore the doctor has the patient take a pill each
day. Surprisingly, a 200 mg pill each day is far too large a dose to maintain a 200
mg level in the bloodstream. If the doctor prescribes 200 mg each day, the patient
will have 200 mg in the bloodstream on the first day. At the end of one day, only
120 mg will remain, but another 200 mg will be added, making the total amount
320 mg. This overdose can potentially be very harmful for the patient, so the doc-
tor needs to determine an ideal dosage that will allow only 200 mg to remain in
the bloodstream at any given time. 
A pharmacist can model this situation by using a spreadsheet or table of val-
ues, making sure that the amount in the bloodstream at the end of the day is 60
percent the amount at the beginning of the day, and then adding that value to the
amount at the beginning of the next day. The following table illustrates how
much of the drug would remain in the bloodstream during the first twenty days
if 200 mg were taken each day. Notice that eventually the amount of drug in the
bloodstream will level off near 500 mg after about ten days.