The Fibonacci numbers are generated by setting
F_{0}=0, F_{1}=1, and then using the
recursive formula
F_{n} = F_{n1} + F_{n2}
to get the rest. Thus the sequence begins:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This sequence of Fibonacci numbers arises
all over mathematics and also in nature.
However, if I wanted the 100th term of this sequence,
it would take
lots of intermediate calculations with the recursive
formula to get a result. Is
there an easier way?
Yes, there is an exact formula for the nth term!
It is:
a_{n} = [ Phi^{n}
 (phi)^{n} ]/Sqrt[5].
where Phi=(1+Sqrt[5])/2 is the socalled golden mean,
and phi=(1Sqrt[5])/2 is an
associated golden number, also equal to (1/Phi).
This formula is attributed to Binet in 1843,
though known by Euler before him.
The Math Behind the Fact:
The formula can be proved by induction. It can also be proved using the eigenvalues of a 2x2matrix that encodes the recurrence. You can
learn more about recurrence formulas in a fun course called
discrete mathematics.
How to Cite this Page:
Su, Francis E., et al. "Fibonacci Number Formula."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

