Assignment # 12
Exploring Fibonacci Sequences
by Michael Ferra
Proposed Investigation
Using a Spreadsheet:
i. Generate a Fibonacci sequence in the first column using f(0) = 1, f(1) = 1, f(n) = f(n - 1) + f(n - 2).
ii. Construct the ratio of each pair of adjacent terms in the Fibonacci sequence. What happens as n increases? What about the ratio of every second term? etc.
iii. Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0) = 1 and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.
i. Generate a Fibonacci sequence in the first column using f(0) = 1, f(1) = 1, f(n) = f(n - 1) + f(n - 2).
Below is an Excel Spreadsheet image showing the first 30 numbers in the Fibonacci sequence where f(n) represents the nth Fibonacci number.
ii. Construct the ratio of each pair of adjacent terms in the Fibonacci sequence. What happens as n increases? What about the ratio of every second term? etc.
Below is an Excel Spreadsheet image showing the first 30 numbers in the Fibonacci sequence where f(n) represents the nth Fibonacci number and f(n + 1)/f(n) represents the ratio of each pair of adjacent terms in the Fibonacci sequence.
What happens as n increases?
Notice that as n increases, the ratio of adjacent terms in the Fibonacci sequence approaches 1.618033989. In other words, the ratio seems to be converging to 1.618033989. Why this number? Does it hold any significance?
Let's denote R(n) as the nth number in the sequence of Fibonacci ratios. Then:
Let's manipulate R(n) a little. We know the Fibonacci Sequence is defined by
Now let's rewrite R(n).
This real number L must then satisfy the equation,
What about the ratio of every second term?
Below is an Excel Spreadsheet image showing the first 30 numbers in the Fibonacci sequence where f(n) represents the nth Fibonacci number and f(n + 1)/f(n) represents the ratio of each pair of adjacent terms in the Fibonacci sequence and f(n + 2)/f(n) represents the ratio of every second term in the Fibonacci Sequence.
Notice that as n increases, the ratio of every second term in the Fibonacci sequence approaches 2.618033989. In other words, the ratio seems to be converging to 2.618033989. Let's explore this.
Let's denote S(n) as the nth number in the sequence of the ratio of every second term in the Fibonacci Sequence. Then:
We've now shown that the ratio of every second term in the Fibonacci Sequence converges to 𝞅 + 1 which is the same as 𝞅2, the square of the Golden Ratio.
iii. Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0) = 1 and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.
Let's explore a sequence where f(0) = 8, f(1) = 3, f(n) = f(n - 1) + f(n - 2).
Notice we have the same pattern of behavior. As n increases, the ratio of adjacent terms in the Fibonacci sequence converges to the Golden mean and the ratio of every second term in the Fibonacci Sequence converges to the square of the Golden Ratio.
Let's generate another sequence f(0) = 50, f(1) = 88, f(n) = f(n - 1) + f(n - 2).
Once again notice we have the same pattern of behavior. As n increases, the ratio of adjacent terms in the Fibonacci sequence converges to the Golden mean and the ratio of every second term in the Fibonacci Sequence converges to the square of the Golden Ratio.
Let's explore the Lucas Sequence where f(0) = 1, f(1) = 3, f(n) = f(n - 1) + f(n - 2).
Amazing!! No matter what values we choose for f(0) and f(1), the ratios still converge to factors of the Golden Ratio!
