Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 10 The Mathematics of Money 10.1Percentages 10.2Simple Interest 10.3 Compound Interest 10.4Geometric Sequences 10.5Deferred Annuities: Planned Savings for the Future 10.6Installment Loans: The Cost of Financing the Present

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. A geometric sequence starts with an initial term P and from then on every term in the sequence is obtained by multiplying the preceding term by the same constant c: The second term equals the first term times c, the third term equals the second term times c, and so on. The number c is called the common ratio of the geometric sequence. Geometric Sequence

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 5, 10, 20, 40, 80,... The above is a geometric sequence with initial term 5 and common ratio c = 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 an increasing sequence: Every term is bigger than the preceding term. This will happen every time the common ratio c is bigger than 1. Example 10.17Some Simple Geometric Sequences

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The above is a geometric sequence with initial term 27 and common ratio Notice that this is a decreasing sequence, a consequence of the common ratio being between 0 and 1. Example 10.17Some Simple Geometric Sequences

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The above is a geometric sequence with initial term 27 and common ratio Notice that this sequence alternates between positive and negative terms, a consequence of the common ratio being a negative number. Example 10.17Some Simple Geometric Sequences

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. A generic geometric sequence with initial term P and common ratio c can be written in the form P, cP, c 2 P, c 3 P, c 4 P,... We will use a common letter–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, G 0 = P, G 1 = cP, G 2 = c 2 P, G 3 = c 3 P, … Generic Geometric Sequence

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. G N = cG N–1 ; G 0 = P (recursive formula) G N = C N P (explicit formula) GEOMETRIC SEQUENCE

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Consider the geometric sequence with initial term P = 5000 and common ratio c = The first few terms of this sequence are G 0 = 5000, G 1 = (1.06)5000 = 5300, G 2 = (1.06) = 5618, G 3 = (1.06) = Example 10.18A Familiar Geometric Sequence

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. If we put dollar signs in front of these numbers, we get the principal and the balances over the first three years on an investment with a principal of $5000 and with an APR of 6% compounded annually. These numbers might look familiar to you–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. Example 10.18A Familiar Geometric Sequence

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Example illustrates the important role that geometric sequences play in 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 in the account at the end of each compounding period are the terms of a geometric sequence with initial term P and common ratio (1 + p): P,  P(1 + p),  P(1 + p) 2,  P(1 + p) 3,... Compound Interest

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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? How long will it take to eradicate the virus? Because the number of reported cases of the gamma virus decreases by 70% each year, Example 10.19Eradicating the Gamma Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. we can model this number by a geometric sequence with common ratio c = 0.3 (a 70% decrease means that the number of reported cases is 30% of what it was the preceding year). We will start the count in 2008 with the initial term G 0 = P = 1,000,000 reported cases. In 2009 the numbers will drop to G 1 = 300,000 reported cases, in 2010 the numbers will drop further to G 2 = 90,000 reported cases, and so on. Example 10.19Eradicating the Gamma Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. By the year 2014 we will be in the sixth iteration of this process, and thus the number of reported cases of the gamma virus will be G 6 =(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 to about 20 cases, and by 2018 to about 6 cases. Example 10.19Eradicating the Gamma Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We will now discuss a very important and useful formula–the geometric sum formula– that allows us to add a large number of terms in a geometric sequence without having to add the terms one by one. Geometric Sum Formula

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. THE GEOMETRIC SUM FORMULA

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. At the emerging stages, the spread of many infectious diseases–such as HIV and the West Nile virus–often follows a geometric sequence. Let’s consider the case of an imaginary infectious disease called the X-virus, for which no vaccine is known. The first appearance of the X-virus occurred in 2008 (year 0), when 5000 cases of the disease were recorded in the United States. Example 10.20Tracking the Spread of a Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Epidemiologists estimate that until a vaccine is developed, the virus will spread at a 40% annual rate of growth, and it is expected that it will take at least 10 years until an effective vaccine becomes available. Under these assumptions, how many estimated cases of the X-virus will occur in the United States over the 10-year period from 2008 to 2017? We can track the spread of the virus by looking at the number of new cases of the Example 10.20Tracking the Spread of a Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. virus reported each year. These numbers are given by a geometric sequence with P = 5000 and 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 Example 10.20Tracking the Spread of a Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. It follows that the total number of cases over the 10-year period is given by the sum (1.4)  (1.4) 2  … + (1.4) 9  5000 Using the geometric sum formula, this sum (rounded to the nearest whole number) equals Example 10.20Tracking the Spread of a Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Our computation shows that about 350,000 people will contract the X-virus over the 10- year period. What would happen if, due to budgetary or technical problems, it takes 15 years to develop a vaccine? All we have to do is change N to 15 in the geometric sum formula: Example 10.20Tracking the Spread of a Virus

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. These are sobering numbers: The geometric sum formula predicts that if the development of the vaccine is delayed for an extra five years, the number of cases of X-virus cases would grow from 350,000 to almost 2 million! Example 10.20Tracking the Spread of a Virus