The Fibonacci numbers are generated by setting F_{0} = 0, F_{1 }= 1, and then using the recursive formula

F_{n} = F_{n-1} + F_{n-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:

a_{n} = [Phi^{n} – (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-matrix 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*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by: **

Arthur Benjamin