Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

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.

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

Share

Keywords:    proof by induction, combinatorics, Binet's formula
Subjects:    combinatorics, number theory
Level:    Easy
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
3.95
 
current
rating 		Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!