Exploring The Fibonacci Sequence

Colleen Foy

The Fibonacci Sequence is an infinite sequence whose first two terms are 1, and the rest of the terms are formed by adding the two previous terms together. Note the spreadsheet below that lists a few terms of the Fibonacci Sequence.

Now let's take a look at the ratio between consecutive terms in the Fibonacci Sequence. We are looking at this ratio: . Here is a spreadsheet of the terms of the Fibonacci Sequence and the ratio between them.

Note that as

ngets larger and larger, the ratio approaches 1.61803399. This ratio/number is known as Phi . Some other names for it include The Golden Ratio and The Golden Number. Now what is the ratio of every second term asngets larger and larger? Here is a spreadsheet that displays and the ratio of every second term.This ratio appears to be . Can we prove this?

Note that when we take the limit we obtain the following:

Another interesting sequence is the Lucas Sequence. The Lucas sequence is formed the same way as the Fibonacci Sequence except the first two terms are 1 and 3 respectively. Let's take a look at the Lucas Sequence in the spreadsheet below.

What is the ratio between the consecutive terms of the Lucas Sequence?

The ratio is still .

Click here for a link to an excel sheet for further exploration.