Fibonacci Number Formula

The 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  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 2×2- 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. “ Number Formula.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Arthur Benjamin

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