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  1987198819891990199119921993199419951996
  Size  11235813213455

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
  1. 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.
  2. Find a polynomial p(x) such that p(x)f(x)=x.
  3. Find the power series expansion for 1 / ax+b and determine its radius of convergence.
  4. Find the partial fraction decomposition for x / p(x) and use it to determine the coefficients for f(x).
  5. What is the radius of convergence for f(x)?
  6. Use your result from 4. to determine a formula for ak 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.


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: