10 The Mathematics of Money

2. 10 The Mathematics of Money 10.1 Percentages 10.2 Simple Interest 10.3 Compound Interest 10.4 Geometric Sequences 10.5 Deferred Annuities: Planned Savings for the Future 10.6 Installment Loans: The Cost of Financing the Present

3. Geometric Sequence A geometric sequence starts with an initial term P and from then on everyterm in the sequence is obtained by multiplying the preceding term by the sameconstant c: The second term equals the first term times c, the third term equalsthe second term times c, and so on. The number c is called the common ratio ofthe geometric sequence.

4. Example 10.17 Some Simple Geometric Sequences 5, 10, 20, 40, 80, . . . The above is a geometric sequence with initial term 5 and common ratioc = 2.Notice that since the initial term and the common ratio are both positive, every term of the sequence will be positive. Also notice that the sequence is anincreasing sequence: Every term is bigger than the preceding term. This willhappen every time the common ratio c is bigger than 1.

5. Example 10.17 Some Simple Geometric Sequences The above is a geometric sequence with initial term 27 and common ratioNotice that this is a decreasing sequence, a consequence of the commonratio being between 0 and 1.

6. Example 10.17 Some Simple Geometric Sequences The above is a geometric sequence with initial term 27 and common ratioNotice that this sequence alternates between positive and negative terms, a consequence of the commonratio being a negative number.

7. Generic Geometric Sequence A generic geometric sequence with initial term P and common ratio c can bewritten in the form P, cP, c2P, c3P, c4P, . . . We will use a commonletter–in this case,G for geometric–to label the terms of a generic geometric sequence, with subscripts conveniently chosen to start at 0. In other words, G0 = P, G1 = cP, G2 = c2P, G3 = c3P, …

8. GEOMETRIC SEQUENCE GN = cGN–1; G0 = P (recursive formula) GN = CNP (explicit formula)

9. Example 10.18 A Familiar Geometric Sequence Consider the geometric sequence with initial term P = 5000and common ratioc = 1.06.The first few terms of this sequence are G0 = 5000, G1 = (1.06)5000 = 5300, G2 = (1.06)25000 = 5618, G3 = (1.06)35000 = 5955.08

10. Example 10.18 A Familiar Geometric Sequence If we put dollar signs in front of these numbers, we get the principal and thebalances over the first three years on an investment with a principal of $5000 andwith an APR of 6% compounded annually. These numbers might look familiar toyou–they come from Uncle Nick’s trust fund example (Example 10.10). In fact,the Nth term of the above geometric sequence (rounded to two decimal places)will give the balance in the trust fund on your Nth birthday.

11. Compound Interest Example 10.18 illustrates the important role that geometric sequences playin the world of finance. If you look at the chronology of a compound interest account started with a principal of P and a periodic interest rate p, the balances inthe account at the end of each compounding period are the terms of a geometricsequence with initial term P and common ratio (1 + p): P, P(1 + p), P(1 + p)2, P(1 +p)3, . . .

12. Example 10.19 Eradicating the Gamma Virus Thanks to improved vaccines and good public health policy, the number of reported cases of the gamma virus has been dropping by 70% a year since 2008,when there were 1 million reported cases of the virus. If the present rate continues, how many reported cases of the virus can we predict by the year 2014? Howlong will it take to eradicate the virus?Because the number of reported cases of the gamma virus decreases by 70%each year,

13. Example 10.19 Eradicating the Gamma Virus we can model this number by a geometric sequence with commonratio c = 0.3(a 70% decrease means that the number of reported cases is 30% ofwhat it was the preceding year).We will start the count in 2008 with the initial term G0= P = 1,000,000 reported cases. In 2009 the numbers will drop to G1= 300,000 reported cases, in2010 the numbers will drop further to G2= 90,000 reported cases, and so on.

14. Example 10.19 Eradicating the Gamma Virus By the year 2014 we will be in the sixth iteration of this process, and thusthe number of reported cases of the gamma virus will be G6=(0.3)6 1,000,000.By 2015 this number will drop to about 219 cases (0.3  729 = 218.7), by 2016 to about 66 cases (0.3  219 = 65.7), by 2017 toabout 20 cases, and by 2018 to about 6 cases.

15. Geometric Sum Formula We will now discuss a very important and useful formula–the geometric sumformula–that allows us to add a large number of terms in a geometric sequencewithout having to add the terms one by one.

16. THE GEOMETRIC SUM FORMULA

17. Example 10.20 Tracking the Spread of a Virus At the emerging stages, the spread of many infectious diseases–such as HIVand the West Nile virus–often follows a geometric sequence. Let’s consider thecase of an imaginary infectious disease called theX-virus, for which no vaccine isknown. The first appearance of the X-virus occurred in 2008 (year 0), when 5000cases of the disease were recorded in the United States.

18. Example 10.20 Tracking the Spread of a Virus Epidemiologists estimate that until a vaccine is developed, the virus will spread at a 40% annual rateof growth, and it is expected that it will take at least 10 years until an effectivevaccine becomes available. Under these assumptions, how many estimated casesof the X-virus will occur in the United States over the 10-year period from 2008to 2017? We can track the spread of the virus by looking at the number of new cases ofthe

19. Example 10.20 Tracking the Spread of a Virus virus reported each year. These numbers are given by a geometric sequencewith P = 5000and common ratio c = 1.4 (40% annual growth): 5000 cases in 2008 (1.4)  5000 = 7000 new cases in 2009 (1.4)2 5000 = 9800 new cases in 2010 … (1.4)9 5000 = 103,305 new cases in 2017

20. Example 10.20 Tracking the Spread of a Virus It follows that the total number of cases over the 10-year period is given by the sum 5000 + (1.4)  5000 + (1.4)2 5000 + … + (1.4)9 5000 Using the geometric sum formula, this sum (rounded to the nearest whole number)equals

21. Example 10.20 Tracking the Spread of a Virus Our computation shows that about 350,000 people will contract the X-virusover the 10-year period. What would happen if, due to budgetary or technicalproblems, it takes 15 years to develop a vaccine? All we have to do is change N to15 in the geometric sum formula:

22. Example 10.20 Tracking the Spread of a Virus These are sobering numbers: The geometric sum formula predicts that if thedevelopment of the vaccine is delayed for an extra five years, the number of casesof X-virus cases would grow from 350,000 to almost 2 million!