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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
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 10.4 - 2
Geometric Sequence
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.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 10.4 - 3
Example 10.17 Some Simple
Geometric Sequences
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.
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Excursions in Modern Mathematics, 7e: 10.4 - 4
Example 10.17 Some Simple
Geometric Sequences
1 1
27,9,3,1, , ,K
3 9
The above is a geometric sequence with
1
initial term 27 and common ratio c  .
3
Notice that this is a decreasing sequence, a
consequence of the common ratio being
between 0 and 1.
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Excursions in Modern Mathematics, 7e: 10.4 - 5
Example 10.17 Some Simple
Geometric Sequences
1 1
27, 9,3, 1, , ,K
3 9
The above is a geometric sequence with
1
initial term 27 and common ratio c   .
3
Notice that this sequence alternates between
positive and negative terms, a consequence
of the common ratio being a negative number.
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Excursions in Modern Mathematics, 7e: 10.4 - 6
Generic Geometric Sequence
A generic geometric sequence with initial
term P and common ratio c can be written in
the form P, cP, c2P, c3P, c4P, . . .
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,
G0 = P, G1 = cP, G2 = c2P, G3 = c3P, …
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Excursions in Modern Mathematics, 7e: 10.4 - 7
GEOMETRIC SEQUENCE
GN = cGN–1 ; G0 = P (recursive formula)
GN = CNP (explicit formula)
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Excursions in Modern Mathematics, 7e: 10.4 - 8
Example 10.18 A Familiar Geometric
Sequence
Consider the geometric sequence with initial
term P = 5000 and common ratio c = 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
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Excursions in Modern Mathematics, 7e: 10.4 - 9
Example 10.18 A Familiar Geometric
Sequence
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.
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Excursions in Modern Mathematics, 7e: 10.4 - 10
Compound Interest
Example 10.18 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, . . .
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Excursions in Modern Mathematics, 7e: 10.4 - 11
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? How
long will it take to eradicate the virus?
Because the number of reported cases of the
gamma virus decreases by 70% each year,
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 10.4 - 12
Example 10.19 Eradicating the
Gamma Virus
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 G0 = P = 1,000,000 reported
cases. In 2009 the numbers will drop to G1 =
300,000 reported cases, in 2010 the numbers
will drop further to G2 = 90,000 reported
cases, and so on.
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Excursions in Modern Mathematics, 7e: 10.4 - 13
Example 10.19 Eradicating the
Gamma Virus
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
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 to about
20 cases, and by 2018 to about 6 cases.
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Excursions in Modern Mathematics, 7e: 10.4 - 14
Geometric Sum Formula
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.
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Excursions in Modern Mathematics, 7e: 10.4 - 15
THE GEOMETRIC SUM FORMULA
N

c  1
2
N1
P  cP  c P  L  c P  P 

c

1


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Excursions in Modern Mathematics, 7e: 10.4 - 16
Example 10.20 Tracking the Spread of a
Virus
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.
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Excursions in Modern Mathematics, 7e: 10.4 - 17
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 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
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 10.4 - 18
Example 10.20 Tracking the Spread of a
Virus
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
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Excursions in Modern Mathematics, 7e: 10.4 - 19
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
10
1.4  1
5000 
 349,068
1.4  1
 
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Excursions in Modern Mathematics, 7e: 10.4 - 20
Example 10.20 Tracking the Spread of a
Virus
Our computation shows that about 350,000
people will contract the X-virus over the 10year 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:
15
1.4  1
5000 
 1,932,101
1.4  1
 
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Excursions in Modern Mathematics, 7e: 10.4 - 21
Example 10.20 Tracking the Spread of a
Virus
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!
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Excursions in Modern Mathematics, 7e: 10.4 - 22