Subject: Fibonacci Numbers
Sender: Gary Nelb

I'm doing a project on fibonacci numbers and I'm using different starting values and finding out if different starting values to see whether or not the ratios still get closer to phi. I was wondering, what numbers should I use. Should I use two of the same # like 2 and 2, or numbers like 1 and 2, or even something totally different. I need your help!

Hi Gary,

THe fibonacci recurrence is f(n) -f(n-1)-f(n-2) = 0. With it we associate its characteristic equation, x2 -x - 1 = 0. The roots are (1+SQRT(5))/2 and (1-SQRT(5))/2. This tells us that f(n) looks like

f(n) = A((1+SQRT(5))/2)n + B((1-SQRT(5))/2)n,

for suitable constants A and B which depend upon the initial conditions. Thus it doesn't matter what numbers that you start with initially, the ratios will always tend to phi = (1+SQRT(5))/2.


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