Assignment #12

by

Megan Dickerson

The Fibonacci Sequence

First in this exploration I created an excel file for the Fibonacci Sequence.

The first column holds the first 43 numbers of the Fibonacci Sequence starting with 1,1,2,.... (f(0) = 1 and f(1) = 1)

The second column looks at A1/A2. This ratio converges to .61803399.

The Third column looks at A2/A1. This ratio converges to 1.61803399.

Both of these ratios are of the adjacent terms, just in opposite order. It is clear that as n increases the ratio converges to a specific number. The graphs below also show this to be true:

The fourth column looks at the ratio of every second term (A3/A1). This ratio converges to 2.61803399.

The fifth column looks at the ratio of every third term (A4/A1). This ratio converges to 4.23606798.

The sixth column looks at the ratio of every fourth term (A5/A1), and it converges to 6.85410197.

I was interested to see how these ratios played out when the first terms of the sequence were not 1,1,2. Thus, I began to look at what is known as a "Lucas Sequence."

A Lucas Sequence is defined as sequences where f(0) and f(1) are some arbitrary integers other than 1.

First I looked at the Lucas Sequence where f(0) = 1 and f(1) = 3.

The first column show the first 43 terms of the sequence, The second column gives the ratio for the adjacent terms (A2/A1). It converges to 1.61803399.

The second column is the ratio of every second term (A3/A1), and it converges to 2.61803399.

The third column is the ratio of every third term (A4/A1), and it converges to 4.23606798.

The fourth column is the ratio of every fourth term (A5/A1), and it converges to 6.85410197.

Even though these number do in fact match the ratios for the original Fibonacci Sequence I was not quite convinced. So I decided to try another Lucas Sequence this time with the first 2 numbers farther apart. I let f(0) = -5 and f(1) = 16.

The first column shows the first 43 terms of this sequence and columns 2-4 represent the same ratios as the last Lucas Sequence.

The ratio for adjacent terms converges to 1.61803399.

The ratio for every second term converges to 2.61803399.

The ratio for every third term converges to 4.23606798.

The ratio of every fourth term converges to 6.85410197.

I then concluded that these ratios would converge to these numbers regardless of f(0) and f(1). However, I just provided examples where this holds, not a formal proof.

The graphs of my last Lucas sequence are below:

 every adjacent term every second term every third term every fourth term