Wiggets and Fibonacci

29 Feb 2002

Mesquite Trees, Anon.
Thorn Avenue
Lubbock, TX 79414

Dear Calculus Student,

I am writing to ask for your assistance in a matter of great importance to our social club.

We at Midgets United have kept careful track of our membership growth over the last 15 years. (Midgets was founded in 1987 with one original member.) Progress has been slow but steady. In order to plan for the future, we need to be able to predict what our membership will be -- say at our centential celebration.

Amazingly, we discovered a pattern in our membership records! Our records show that our membership growth follows the Fibonacci pattern. Even if you haven't seen them before you may recognize the pattern. To get each of the numbers after the second one you simply add the two previous numbers. That is exactly how our membership has progressed over our first 10 years.

Year	1987	1988	1989	1990	1991	1992	1993	1994	1995	1996
Size	1	1	2	3	5	8	13	21	34	55

The amazing thing is that according to Nature Magazine our pattern occurs almost universally in nature -- growth of spirals on sea shells and pine cone, growth of rabbit colonies (that last is disgusting to think that our club would be compared to rabbits, conies, whatever.)

We want to be able to present at our next club meeting our exciting discovery. More specifically we want to be able to show our members a formula which will allow us to predict our future. Think of it -- our centennial club, our bicentential club, our millenial club!!! But no one here knows how to or wants to calculate it -- yuck.

In Nature Magazine there was a scheme for calculating Fibonacci numbers, but it didn't make sense to any of us. Will you take a look at the directions given there and figure it out for us, please.

Directions from Nature Magazine for Calculating Fibonacci Numbers

Let a₁, a₂, a₃,. . . represent the Fibonacci sequence. Consider the power series f(x) = a₁x + a₂x² + a₃x³ + . . . . Show that the radius of convergence of this power series is at least 1/2.
Find a polynomial p(x) such that p(x)f(x)=x.
Find the power series expansion for 1 / ax+b and determine its radius of convergence.
Find the partial fraction decomposition for x / p(x) and use it to determine the coefficients for f(x).
What is the radius of convergence for f(x)?
Use your result from 4. to determine a formula for a_k for each value of k. Use a calculator or computer to verify that the formula you derived is correct for small values of k.

Could you find this formula for us and explain the directions to us? Especially, could you find for us our future our centennial, bicentential and millenial club membership sizes?

Our next meeting will be in late April. If you could give us a report back by 23 April 2002 at 5:00 pm, then we could present it to our members. Thanks in advance.

Sincerely,

Johnny Longfellow

Recording Secretary
Midgets United

cc: Dr. Kent Pearce

Technical Report Requirements

All reports submitted to Midgets United should be written so that the directors receiving the report can understand and apply the information contained therein. Owing to Midgets United preeminent position all of our officers have degrees in agricultural sciences, and thus have had college level mathematics, including calculus---unfortunately, however, their long experience in the field precludes a ready knowledge of the same. Therefore, the reports should assume a strong precalculus and basic calculus (about half a semester of calculus I) background, but should not expect a knowledge of much more than that.

Reports should further:

Be written in the first person plural (e.g., "We found the requisite data from the figure...").
Include mathematical formulas and appropriate graphs in the body of the report as appropriate to describe the methods and results obtained. (While the report must be typewritten, it is fine to neatly hand-write formulas if that significantly simplifies its generation.)
Clearly explain how the mathematical formulas that are included bear on the problem being solved.
Consist of:
- An Introduction, describing the problem to be solved, and an indication of the mathematical method used to solve it.
- A Body, describing the mathematical problem that was solved to answer the question(s) posed in the introduction, and the solution to it.
- A Conclusion, summarizing the results obtained from the solution described in the body and clearly stating their relevance to the original problem as described in the introduction.
- Be 2.5--5 pages in length, excluding supporting figures and diagrams in an attached appendix.