# Fibonacci Numbers

The Fibonacci numbers are defined by the following recursive formula:  Thus, each number in the sequence (after the first two) is the sum of the previous two numbers.

(Some people start numbering the terms at 1, so , , and so on. But the recursion is the same.)

The first few Fibonacci numbers are: Fibonacci numbers have been extensively studied. Koshy  and Rao  have extensive lists of Fibonacci identities; Koshy also has many applications. The Fibonacci Quarterly is a journal devoted to Fibonacci numbers and related topics.

Example. Express each of the following as a single Fibonacci number.

(a) .

(b) .

(c) .

(a) The number after and is , so (b) Since , (c) Example. Prove that if , then  Many results about Fibonacci numbers can be proved by induction.

Example. Prove that For , the left side is and the right side is The result is true for .

Suppose the result holds for n: I'll prove it for . This proves the result for , so the result is true for all by induction. Example. Prove that for , For , the left side is The right side is The result is true for .

Assume the result for n: Prove the result for : This proves the result for , so it's true for by induction. Example. ( An explicit formula for the Fibonacci numbers)

(a) Let Prove: (b) Prove that (a)  For the third and fourth equations, note that and are roots of the quadratic equation So:  (b) For , I have . The right side of the equation above becomes The result is true for .

For , I have . The right side of the equation above becomes  The result is true for .

Assume that the result is true for all . In particular,  I'll prove the result for .   This proves the result for , so the result is true for all by induction. Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York: John Wiley and Sons, 2001.

 K. Subba Rao, Some properties of Fibonacci numbers, American Mathematical Monthly, (10) 60 (1953), 680--684.

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