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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Fibonacci Number Formula

The Fibonacci numbers are generated by setting F0=0, F1=1, and then using the recursive formula

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! It is:

an = [ Phin - (phi)n ]/Sqrt[5].

where Phi=(1+Sqrt[5])/2 is the so-called golden mean, and phi=(1-Sqrt[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 2x2-matrix that encodes the recurrence. You can learn more about recurrence formulas in a fun course called discrete mathematics.

Su, Francis E., et al. "Fibonacci Number Formula." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    proof by induction, combinatorics, Binet's formula
Subjects:    combinatorics, number theory
Level:    Easy
Fun Fact suggested by:   Arthur Benjamin
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