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Write-up


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 99th or 100th 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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