The Fibonacci Sequence is defined recursively as
There have been many interesting investigations into the Fibonacci sequence, including many natural phenomena displaying patterns involving Fibonacci numbers. In this investigation we will begin by noticing the pattern when we take the ratio of consecutive terms in the Fibonacci Sequence. Using a spreadsheet, the first 25 terms of the Fibonacci Sequence were generated (column 1), and the ratio of each term to the previous term is calculated (column 2).
Clearly, as we go further along in the terms of the Fibonacci Sequence, the ratios of consecutive terms are approaching a value of about 1.618.
The graph belows shows this as well:
In order to find the exact value of this limit, we notice that the ratio of consecutive terms is given by
So, this shows that, in the limit, the ratio of consective terms of the Fibonacci Sequence is the Golden Ratio. This is a remarkable result!
Each power of the Golden Ratio simplifies to a binomial with coefficients that are consecutive Fibonacci numbers! Again, an amazing result!!
If this pattern continues, we would expect that
We can prove this statement by mathematical induction. But, first let's explore the Fibonacci Sequence in the negative direction.
Returning to our spreadsheet, which lists the Fibonacci Sequence in the first column, and in subsequent columns calculates the ratio of consecutive terms, the ratio of every other term, the ratio of every third term, and so on, we notice a pattern among these ratios:
We have already seen that the ratio of consecutive terms, in the limit as n goes to infinity, is the Golden Ratio φ, as seen in column 2.
Claim: The ratio of every other term (as seen in column 3), in the limit as n goes to infinity, is φ2.
To see this, we notice that the ratio of every other term is given by
Continuing in this manner, we see that the ratio of every third term (as seen in column 4), is given by
Similarly, the ratio of every fourth term (as seen in column 5), is given by
In general, the ratio of every rth term, in the limit as n goes to infinity, is φr. Stated more formally,
Looking at the columns in the spreadsheet below, where the first column lists the Fibonacci Sequence in the negative direction, we see that these ratios follow the same pattern as previously seen, with the exception of the ratios in each column alternating between positive and negative values:
For each ratio of every rth term (for example, the 4th column shows the ratio of every third term),
There are two interesting relationships that come from the powers of φ. First,
This is clear, since
It is clear that the ratio of consecutive powers of φ is φ, but it is also true (as seen above) that
Because of these two relationships, a rectangle constructed such that the lengths of its sides are consecutive powers of φ will be a Golden Rectangle! Moreover, this process can continue to infinity, as seen in the figure below:
Click here to see a video of the construction.
For one of these Golden Rectangles, whose dimensions are consecutive powers of φ, let us find the length of the diagonal. We start with the first such Golden Rectangle:
The length of its diagonal is given by:
Next, we'll look at the second such Golden Rectangle:
The length of its diagonal is given by:
The table below shows the development of the pattern:
If the pattern continues, we would expect that
This is clear, since
