# The Fibonacci numbers and the golden ratio

Sofie Bronée 1v.         Fibonacci project.         11/04/08

The Fibonacci Numbers and the Golden Ratio

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The Fibonacci Numbers

The Fibonacci numbers are sequence of numbers. They are named after the Italian mathematician Leonardo of Pisa, known as Fibonacci. He published a book called “Liber Abaci”, and he was the first person to publish a book in Western Europe that used the Indian numerals 9, 8, 7, 6, 5, 4, 3, 2, 1. Fibonacci was perhaps the greatest mathematician of his time. But he is most famous for the numbers which has his name.

These are the first ten terms in the Fibonacci sequence.

To get the next term in the sequence the two previous terms must be added. This can be written as:

Un = Un-1 + Un-2

A number in the sequence is the sum of its two predecessors.

The Golden Ratio

To find the Golden Ratio, a term has to be divided with the previous term

(i.e. . .. etc). By doing this a certain amount of times you’ll reach the Golden Ratio. It can be done, no matter what the starting numbers are. To prove my assumption I made several different tables and graphs that shows the Golden Ratio. I’ve used different numbers; negative and positive (look in appendix 1).

These are the “original” start numbers 1 and 1.

By investigating the Fibonacci numbers, I can make a conjecture, that the ratio of two consecutive terms gets closer, as they increase, to the Golden Ratio. By changing the start numbers I proved that it doesn’t make a difference which numbers we use.

To prove my conjecture I’ve changed the equation and solved it. Afterwards I found the discriminant (5), which leads me to find the roots. I plugged in the discriminant to the quadratic equation for finding the roots. The results ...