The Fibonacci Sequence and The Golden Ratio

by:

Brandt Hacker

The Fibonacci sequence is a sequence that begins with  and  set equal to 1.  To find any term in the sequence after  and  we simply add the two previous terms in the sequence.  The rule for the nth term of this sequence can be written as .

While calculating the Fibonacci sequence is basic enough to be done with paper and pencil, the use of a spreadsheet can allow these tedious calculations to be done in an instant.  In the captured spreadsheet below, we have three columns:  the first telling us what term in the sequence we are looking at, the second giving the corresponding number to the term, and the third giving the ratio of the current term to the previous term.

While the ratio begins by oscillating back and forth, it appears to be approaching a fixed number as the value of n increases.  The value that we see in the ratio column is getting closer and closer to the golden number as we move down the page.  The golden number is the ratio at which .

The next aspect of the Fibonacci sequence that we will explore, is what happens if we change the beginning numbers of the sequence?  When we change the first two numbers we change the sequence from a Fibonacci sequence to what is called a Lucas sequence.  The numbers will undoubtedly change, but what about the ratio of each term to its previous term?  Below is an example of a Lucas sequence in which the first term is 3 and the second term is 11.

Above we see that the ratio is still approaching the golden number as n increases.

Lastly, we will look at what happens when we compare the ratios of various terms of the Fibonacci sequence.  The spreadsheet showing this data is seen below:

At first glance it may not appear that these ratios have much to do with one another.  However, as we analyze the columns we see that the ratio of  is equal 1 plus the ratio of .  What about the third column though?  How is  related to the other two, or is it?  There is in fact a relationship; the relationship is that each successive ratio is the previous ratio multiplied by the golden number.  In other words,  = 1.618,  = ,  = , etc.