The Fibonacci sequence is a recursive sequence, defined by

The sequence can then be written as

## Properties

where is the golden ratio.

### Proof

We are given this recurrence relation,

Which is subject to and . One may form an auxiliary equation in accordingly and solve for .

Through the use of the quadratic formula, one will obtain,

or equivalently

So we have,

where and are constants to be determined. Substituting the values we have

Solving for both variables, we obtain,

So, one has

as required.

## Sum

For all integers ,

### Proof

Proposition: given

as defined above,

Let ,

Therefore the proposition holds for . Assume that the proposition holds for . We may now make use of the inductive step. Let .

.

We know that

from the assumption that the proposition holds for .

So, we have,

Using the definition,

one obtains

which obeys the proposition

As the proposition holds for , and , the proposition holds for all natural numbers.

## Binet's Formula

Binet's Formula is a theorem that allows one to determine  , where represents the Fibonacci Number.

The theorem is as follows:

## Trivia

• Fibonacci numbers are claimed to be common in nature; for example, the shell of a nautilus being a Fibonacci spiral. However, this has been disputed with the spiral having a ratio measured between 1.24 to 1.43.[1]

## References

1. Peterson, Ivars (April 1, 2005). "Sea Shell Spirals". Science News.

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