Exploration 12

By: Taylor Adams

The
Fibonacci sequence is represented by the formula

Given
that f(0)=1 and f(1)=1. Therefore, f(2) = f(1)+f(0) = 1+1 = 2.

Using
an Excel sheet, I generated the Fibonacci sequence in column A up to f(100). In column B,
I constructed a ratio, of
the terms in the Fibonacci sequence. The
ratio begins with f(1) and gives us At f(2), we have 2.

The
values in this column fluctuate between 1 and 2 and look like they begin to
approach some number. By f(17), the
ratio has approached 1.618034. The ratio
of every value after this point is essentially this number.

It
is interesting that the ratio of the Fibonacci numbers approach this number,
because this is the golden ratio. The
golden ratio is represented by the formula

If
we set x=,
we get

If
we look at the ratio of ,
as seen in column B of the Excel sheet, the ratio at f(n) holds this relationship
with the ratio before it. For example, the
ratio of f(4)=1.5.

This is also equivalent to

Let’s
see if this relationship carries with x^{2 }and the ratio To do this, let’s first multiply through
by x to get

This
equation tells us that the ratio that the ratio approaches should be the golden ratio squared,
which is also equal to one added to the golden ratio.

If
we look at the ratio of ,
as shown in column C of the Excel sheet, we can see that this relationship is
true. The values, to begin with,
fluctuate between 2 and 3, but end up approaching a limit at 2.618034 by f(18).

We
can visually see this ratio reaching the limit 2.618034 through the following
graph:

When
the golden ratio, x, is cubed, we get the equation…

This
tells us that the values for the ratio should approach.

The ratio is shown in column D of the Excel sheet. We can see that the values for this ratio do approach
4.236068. In fact, they reach this limit
at f(20).

Let’s
see of x^{4 }is also the ratio of . First, let’s multiply x^{3} by x to
get…

This
tells us that the ratio of should approach

If
we look at this ratio shown in column E of the Excel sheet, we can see that
this ratio does approach 6.854102 at f(22).

This
should also be true for the ratio .

This
tells us that the ratio should approach

If
we look at column F, we can see that this ratio does approach 11.09017 at f(21).

Let’s see if this relationship occurs in another sequence. The Lucas sequence, for example, is very similar to the Fibonacci sequence in that

Except
f(0)=1 and f(1)=3.

I
constructed the Lucas sequence in column H and set up the same ratios as I did
for the Fibonacci sequence in columns I-M respectively.

It
is interesting to see that not only does this sequence approach some value for
each ratio, but it also approaches the golden ratio
and holds the same relationships when the golden ratio is raised by some
power. Actually, all sequences like
these approach the golden ratio and have the same limits for the other ratios
as well.