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Lipitor is a member of the drug class known as statins and is used for loweringblood cholesterol. Lipitor is eliminated from the patient’s body at a rate thatis proportional to the amount of the drug that is in the body. Suppose that65% of the drug that is in the body is eliminated within 24 hours.1 Supposethat 40 mg of Lipitor is taken at the same time every morning. We wouldlike to find out how much Lipitor is in a patient’s body after several years oftaking the medication.
Immediately after the first dose of Lipitor, the patient will have 40 mg in his body. After one day 65% of the drug will be eliminated or only 35% ofthe dose will be left in the patient’s body. After the second dose of the drug,the patient will have milligrams of the the drug in his system; however, only 35% of the drug willremain in his system after another 24 hours, or 0.35[40 + 0.35(40)] = (0.35)40 + (0.35)2(40) mg.
The patient will have 40 + (0.35)40 + (0.35)2(40) milligrams of Lipitor in hissystem immediately after taking the third dose of the drug. We can nowbegin to see a pattern.
1This figure is consistent with the actual half-life of Lipitor, which is about 14 hours.
Milligrams of Lipitor in the patient’ssystem immediately after taking the drug 40 + (0.35)40 + (0.35)2(40) + (0.35)3(40) 40 + (0.35)40 + (0.35)2(40) + (0.35)3(40) + · · · + (0.35)n−1(40) After several years, n will become very large. For example, after 3 years, 40 + (0.35)40 + (0.35)2(40) + (0.35)3(40) + · · · + (0.35)1094(40) could prove quite time consuming. Fortunately, we can find an easier way.
A finite geometric sum is a sum of the form where a is the first term in the series and r is the constant ratio of any termto the previous term. In our example, a = 40 and r = 0.35. Let us computethe sum of an arbitrary geometric sum S = a + ar + ar2 + ar3 + · · · + arn.
First, we will multiply both sides of this equation by r to obtain rS = ar + ar2 + ar3 + · · · + arn+1.
S − rS = (a + ar + ar2 + ar3 + · · · + arn) − (ar + ar2 + ar3 + · · · + arn+1).
Notice that all of the terms on the right-side of this equation cancel outexcept for the very first term and the very last term. Thus, We can now compute the amount of Lipitor in a patient’s bloodstream after3 years to be We find that (0.35)1094 is a very small number. Therefore, we can estimatethe amount of Lipitor in the patient’s body by In fact, we might safely assume that the amount of Lipitor in a patient’ssystem would be approximately 61.5 mg if that person had been treatedwith the drug for a very long time.
Finite Geometric Sums. A finite geometric sum is a sum of the form where r is the ratio of any term to the previous term and a is the firstterm. (Notice that we have n + 1 terms in this sum—n is not thenumber of terms in the sum.) The sum of any finite geometric sum is It is now easy to find other examples of geometric sums.
5 + 10 + 20 + 40 + 80 + · · · + 5120 = 5 + 5 · 2 + 5 · 22 + 5 · 23 + · · · + 5 · 210 where a = 2/3, r = −1/3, and n = 14.
Suppose that Congress is considering a tax cut to stimulate the economy.
For example, they might propose a tax cut totaling $1 billion. If we assumethat those receiving the tax cut will spend 70% and save 30%, then $700million would be injected back into the economy. Others would receive this$700 million and would in turn spend 70% of this amount while saving therest. That is, they would spend $700(0.7) million. If this trend continues,then 700 + 700(0.7) + 700(0.7)2 + · · · + 700(0.7)n million dollars would be injected back into the economy. This is just a finitegeometric sum which adds up to Before we take the trouble to evaluate this sum, we should notice that(0.7)n+1 will be very small for very large n. In fact, for arbitrarily large n, we can consider this term to be zero. Hence, the net effect of our tax cutcould be considered to be A infinite geometric series is a sum of the form where a is the first term in the series and r is the constant ratio of any termto the previous term. We can estimate this series by taking larger and largerfinite sums, notice that the term rn+1 gets arbitrarily small provided that −1 < r < 1.
Thus, we can consider the sum of an infinite geometric series to be Infinite Geometric Series. A infinite geometric series is a sum ofthe form where r is the ratio of any term to the previous term and a is the firstterm. The sum of a geometric series is provided that −1 < r < 1. For r ≥ 1 or r ≤ −1, the series does notconverge.
5 + 10 + 20 + 40 + 80 + · · · = 5 + 5 · 2 + 5 · 22 + 5 · 23 + · · · does not converge, since r = 2 is greater than 1.
Geometric series have many applications in mathematics, science, and eco-nomics. The following exercises demonstrate a few of these applications • A middle school science class has performed an experiment in which they have dropped a ball from a height of 2 meters. They have mea-sured that each time it strikes the ground it rebounds to about 3/4 ofits previous height. How far will the ball have traveled before it comesto a complete rest? • A pendulum bob swings through an arc of 40 cm long on its first swing.
Each swing thereafter, it swings only 80% as far as on the previousswing. How far will it swing altogether before coming to a completestop? • Use the theory of geometric series to find the fractional representation • Suppose that the following information is typed into the first two rows and first three columns of your spreadsheet. If row 2 is copied intorows 3 through 15, what information will be in each column in row 15?What will cell C15 tell you? • Suppose that $100,000 of counterfeit money is introduced into the econ- omy. After this initial amount is spent, 25% is detected as fake and isremoved from circulation. In fact, each time the counterfeit money isused, 25% is detected as fake and removed from circulation. Determinethe total amount of counterfeit money spent in all transactions.
Fibonacci was (c. 1170 c. 1250) was an Italian mathematician who was oneof the people responsible for introducing the Hindu-Arabic numeral system to Europe. One of the problem posed by Fibonacci was the following.
Suppose that rabbits live forever and the every month each pair ofrabbits produce a new pair of rabbits which becomes productiveat the age of two months. If we begin with one newborn pair ofrabbits, how many pairs of rabbits will we have in the nth month? If we let fn be the number of pairs after n months, then we know that f1 = 1 and f2 = 1. After two months, we have a productive pair of rabbits,so f3 = 1 + 1 = 2 or f3 = f1 + f2. In the nth month, each pair that is twomonths old or older will produce a new pair to add to the fn−1 pairs alreadypresent, or This sequence of numbers is known as the Fibonacci numbers.2 A sequence is a list of numbers with a definite order. The following areexamples of sequences.
2Fibonacci sequences are interesting to many mathematicians. There is even an aca- demic journal, The Fibonacci Quarterly, devoted to publishing research papers about theFibonacci numbers.
We will denote the sequence a1, a2, a3, . . . , an, . . . by Using this notation, we can write the sequence as {2−n}. Some sequences such as the Fibonacci sequence can only be writtenrecursively, f1 = 1, f2 = 1, fn = fn−1 + fn−2, for n ≥ 2.
A sequence {an} is said to be bounded if there exists a number M such that −M ≤ an ≤ M for all n. A sequence is monotone if the sequence is always in-creasing or always decreasing. For example, the sequence 1, 1, 2, 3, 5, 8, 13, 21, . . .
is monotone increasing while the sequence 1, 0, 1, 0, 1, 0, . . . is bounded butnot monotone.
gets closer and closer to 0 as n gets larger and larger. We say that thissequence converges to 0 or has a limit of 0. On the other hand, the sequences do not have a limit. These sequences are said to diverge.
Limits. A sequence {an} has the limit L if we can make the terms anas close to L as we like by taking n sufficiently large. In this case wewrite and say that the sequence converges. If a sequence does not have alimit, we say it diverges.
One of the key facts about sequences is that a bounded monotone se- quence always converges. for example, the sequence is increasing but never gets larger than 2. Therefore, this sequence mustconverge. To convince yourself that the sequence is indeed bounded, you canrepresent it on a number line.
Monotone Sequence Theorem. Every bounded monotone sequenceconverges.
for any natural number n. This formula is easily verified for small numberssuch as n = 1, 2, 3, or 4, but it is impossible to verify for all natural numberson a case-by-case basis. To prove the formula true in general, a more genericmethod is required.
Suppose we have verified the equation for the first n cases.
attempt to show that we can generate the formula for the (n + 1)th casefrom this knowledge. The formula is true for n = 1 since If we have verified the first n cases, then This is exactly the formula for the (n + 1)th case.
This method of proof is known as mathematical induction. Instead of attempting to verify a statement about some subset S of the positive integers N on a case-by-case basis, an impossible task if S is an infinite set, we give aspecific proof for the smallest integer being considered, followed by a genericargument showing that if the statement holds for a given case, then it mustalso hold for the next case in the sequence. We summarize mathematicalinduction in the following axiom.
Principle of Mathematical Induction. Let S(n) be a statementabout integers for n ∈ N and suppose S(n0) is true for some integer n0.
If for all integers k with k ≥ n0 S(k) implies that S(k + 1) is true, thenS(n) is true for all integers n greater than n0.
For all integers n ≥ 3, 2n > n + 4. Since the statement is true for n0 = 3. Assume that 2k > k + 4 for k ≥ 3. Then2k+1 = 2 · 2k > 2(k + 4). But 2(k + 4) = 2k + 8 > k + 5 = (k + 1) + 4 since k is positive. Hence, by induction, the statement holds for all integersn ≥ 3.
Every integer 10n+1 + 3 · 10n + 5 is divisible by 9 for n ∈ N. For n = 1, is divisible by 9. Suppose that 10k+1 + 3 · 10k + 5 is divisible by 9 for k ≥ 1.
Then 10(k+1)+1 + 3 · 10k+1 + 5 = 10k+2 + 3 · 10k+1 + 50 − 45

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