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 [1] and Rao
[2] 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.

[1] Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York: John Wiley and Sons, 2001.

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

Copyright 2019 by Bruce Ikenaga