## Frequently Asked Questions

#### What are Fibonacci numbers?

Theoretically, cells are able to split and multiply exponentially. Imagine if a tree grew in this manner (the numbers at the side indicate the total number of branches at each stage of growth).

Apart from anything else it doesn't look much like a tree. It is too perfect, too symmetrical. In reality living 'things' take time to mature before they are themselves able to 'split' and if we stagger its growth slightly, that is each 'branch' must grow one period before it is able to split, we obtain a remarkable result.

Although it is still not a tree, it is far more realistic. Leonardo Fibonacci of Pisa (1170-1250), better known as Fibonacci, was the number theorist who brought the Aramaic number system we use today to Europe. He discovered this growth pattern and observed the similarities with breeding pairs of rabbits and other animals. He became aware that rabbits were not able to reproduce immediately and required a maturing period. After this they were able to 'split', in the same manner as the tree above.

Fibonacci was particularly interested in the numbers that are generated. Look carefully at the sequence of numbers 1, 1, 2, 3, 5, 8, 13, ... . You will notice that each new term is the sum of the previous two terms; the next term being 21. This can be expressed using a recurrence relation.

`F`

_{n+2}=

`F`

_{n}+

`F`

_{n+1}

We generally use `F`_{n} to represent the `n`th term in the Fibonacci sequence, as it has become known. It is called a second order recurrence relation, because it requires information about two previous terms.

In fact, the `n`th term, `F`_{n}, is given by the rather cumbersome formula:

F_{n} = |
1 √5 |
1 + √5 2 |
^{n} |
– | 1 – √5 2 |
^{n} |