Latest update to this document:30 October 1995
THIS AREA UNDER CONSTRUCTION
The Fibonacci sequence has been famous for many years. It is defined as follows: F(0)=0, F(1)=1, F(2)=F(1)+F(0), F(3)=F(2)+F(1),...,F(n+1)=F(n)+F(n-1),... Work out and write in a column the Fibonacci numbers through F(20). Then, to the right of F(n), write the value of F(n)/F(n+1) evaluated to seven decimal places. As you work your way down the column, try to guess the value of each entry before you find it with your calculator. Work out the values of (F(n+1)/F(n+2))-(F(n)/F(n+1)) as fractions for the first few values of n. What do you see? Is that consistent with what you noticed as you were writing the decimal values? Use algebra to see if you can prove that what you have noticed will indeed go on as n goes to infinity.
In the last column, the differences between the ratios approaches zero which is consistent with what you can see from the fact that the ratios approach the same number as n increases.
The limit as n approaches infinity of F(n)/F(n+1) is equal to
2/(1+the square root of 5)
which is also equal to .6180340 to seven decimal places.
From the values of the two sets of ratios we can see that this is true:
1+F(n-1)/F(n)=F(n+1)/F(n)
and given that F(n)=F(n+1)-F(n-1) then
1 + F(n-1)/(F(n+1)-F(n-1)) = F(n+1)/(F(n+1)-F(n-1)) and
F(n+1)-F(n-1)+F(n-1)=F(n+1) and
F(n+1)=F(n+1) . This shows that the ratios of sequential values of F(n) are inversely related.
Also notable is the result of column 3, where we divided F(n+1)/F(n). As n approaches infinity, this ratio tends toward the number 1.61803....., which is known as the "Golden Ratio". This number is used in connection with miles/kilometer conversion, as well as in music, architecture, and in connection with stock pricing.
__________________________________________________________________
Info on The Fibonacci Quarterly
Back to Calculus Roster
Back to Northern Michigan University
__________________________________________________________________
Prepared by: Carl, Cancilla, Galbreth, Harris, Hill, Langson
see calc roster for e-mail addresses