Greenwood press

EXPONENTIAL GROWTH
33

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one already in the business sharing some of the franchise fees. In this type of
scheme, millions of dollars can come to the originators, even if none of the prod-
uct is ever sold. The people who pay franchises late in the scheme lose all their
money. When all operations are based on money from new investors rather than
goods or services, the fraud is called a “Ponzi scheme.”
online sources for further exploration
Population changes


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Savings, credit, and compound interest



Inflation rates and calculators



Chain letters and scams





Food technology

Internet growth data

Pricing diamond rings

The US national debt clock

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34
EXPONENTIAL GROWTH

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FIBONACCI SEQUENCE
The infinite sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . is called the Fibonacci
sequence after the Italian mathematician Leonardo of Pisa (ca.1175–ca.1240),
who wrote under the name of Fibonacci. The sequence starts with a pair of ones,
then each number is the sum of the two preceding numbers. The formula for the
sequence is best written recursively (first formula below), rather than the explicit
formula on the right.
Fibonacci established a thought experiment about counts of animals over
generations, and can be described in terms of the family line of honey bees. A
male bee develops from an unfertilized egg—hence has only a mother. Female
bees develop from fertilized eggs; therefore female bees have a father and
mother. How many ancestors does a male bee have? The male bee has one
mother. The mother has a mother and a father. So the male bee has one ancestor
at the parent generation. He has two ancestors at the grandparent generation. If
you work out the great-grandparent generation, you will find that there are three
ancestors. A full picture of the family tree for the bee going back to great-great-
great grandparents will show that the generation counts are 1, 2, 3, 5, 8, 13. 
Placing the male bee at the beginning of the sequence (starting generation)
gives 1, 1, 2, 3, 5, 8, and so on. If you repeat the argument with a female bee, you
will also get a Fibonacci sequence starting with 1, 2, 3, 5, 8, . . . . The sequence
has been shown to have remarkable mathematical properties and some surpris-
ing connections to events outside of mathematics. Eight hundred years after
Fibonacci’s publication of the sequence, an organization and journal, the Fibo-
nacci Quarterly, are devoted to exploring new discoveries about the sequence.
The ratios of consecutive terms of the Fibonacci sequence 

a
n
a
n−1
 produce a 
sequence
1, 2, 1.5, 1.¯
6, 1.6, 1.625, . . . which converges to the golden ratio
1+
√
5
2
≈ 1.61803. If a sequence of squares is built up from two initial unit