The **Fibonacci sequence** is a recursive sequence, defined as

## Properties

is equal to 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

Or otherwise,

- or

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 greater than or equal to ,

### 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.

## 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

- ↑ Peterson, Ivars (April 1, 2005). "Sea Shell Spirals". Science News. https://www.sciencenews.org/article/sea-shell-spirals.

This article is a stub. You can help Math Wiki by expanding it. |