The Fibonacci numbers are generated by setting
F0=0, F1=1, and then using the
Fn = Fn-1 + Fn-2
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 n-th term!
an = [ Phin
- (phi)n ]/Sqrt.
where Phi=(1+Sqrt)/2 is the so-called golden mean,
and phi=(1-Sqrt)/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 2x2-matrix that encodes the recurrence. You can
learn more about recurrence formulas in a fun course called
How to Cite this Page:
Su, Francis E., et al. "Fibonacci Number Formula."
Math Fun Facts.