Title
Multiplying Rabbits
Problem Statement
A newly born rabbit is capable of reproducing at one month old (when it
matures). Suppose the rabbit never dies, and it continues reproducing one
new rabbit every month. So, when the rabbit is born, it has one member in
its own family. After a month, it matures, and by the second month it adds
a new born member to its family. In the third month, the rabbit produces
another offspring; its first child also matures and will be ready to have
an offspring in the next month.
The sequence named by Fibonacci (1,1,2,3,5,8,13,21,...) can describe the
number of members in the rabbit's family at each month. Explain how.


Extensions


Compare the ratio of consecutive numbers in this sequence. For example 1/
1, 1/ 2, 2/ 3, 3/ 5, 5/ 8, .... What do you notice after a while?

Problem setup
This problem requires the use
of the Fibonacci sequence in order to determine the number of rabbits that
are born and mature during a particular month.
Plans to
Solve/Investigate the Problem
My initial plan towards the
solution of this problem was pretty much already laid out in the problem
itself. The problem tells us
that the sequence is named by the Fibonacci sequence. In addition to the Fibonacci
sequence, I planned to draw out a detailed diagram of the new and mature
rabbits born at each month.
Investigation/Exploration
of the Problem
The Fibonacci sequence
describes the members in the rabbit’s family at each month. Therefore, the next term in the
sequence is determined by adding the two previous terms together, which is
how we determined the number of rabbits in the family after n months. I also extended the chart given in
the problem, which allowed me to distinguish between the new and mature
rabbits. We determined, using
an Excel spreadsheet that the sequence could go on forever.
The second part of the problem asked
us to compare the ratio of consecutive numbers in the sequence. Again, we used an Excel spreadsheet
to create a formula that compared each successive ratio based on the two
previous terms in the Fibonacci sequence. After a while, we noticed that the
ratio produced the same result.
I also noticed, by drawing an extension of the chart, that the
numerator indicates the number of mature rabbits in the family.
Extensions of the Problem
Possible extensions of the
problem are to try to determine the 99^{th} or 100^{th}
term in the sequence. We
explored this possibility by using Excel to manipulate various
formulas.
Author & Contact
Mesha Rainey.
mrainey@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
The
Fibonacci Sequence
The Golden Rectangle
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