The following is the results of an exploration in which the Fibonacci sequence was generated using Microsoft Excel. Further investigation was then made such as
1. What type of sequence is obtained if f(0) =1, f(1) =3 or if f(0) = 3 , f(1) = 4, etc...
2. What is the limiting value of the ratio of consecutive terms of such a sequence as mentioned above?
3. What is the limiting value of the ratio of every second term of the Fibonacci Sequence?
For a table showing the Fibonacci Sequence and Ratios of the Consecutive Terms of this sequence, click here.
As you can see, the limiting value of the ratio of consecutive terms of the Fibonacci Sequence seems to approach 1.6180339, a number that has come to be called "the Golden Ratio."
Why does the limiting value of the ratio of consecutive terms of the Fibonacci Sequence approach this special number? Let's see why. . .
As previously stated, the Fibonacci Sequence is defined as: F(n+1) = F(n) + F(n-1).
Therefore, the ratio of consecutive terms of this sequence is defined as:
L,The limiting value of the the ratio of consecutive terms of the Fibonacci Sequence, must be equal to the defined value of the ratio. In other words,
Plugging this back into our equation for the ratio of the consecutive terms, we get
Using the quadratic equation to solve for L, we find that
1. Let F(n) = F(n-1) +F(n-2) where F(0) = 1 and F(1) = 3. This type of sequence is called the Lucas Sequence. The following table shows the first 22 entries of the Lucas Sequence in the left column and the ratio of the consectutive terms of this sequence in the right colum.
For a table showing the first 22 values of the Lucas Sequence as well as Ratios of the Consecutive Terms, click here.
Are you Surprised to see that the limiting value of the ratio of consecutive terms of the Lucas Sequence is also L= 1.61803399?
2. Let's look at another variation of the Fibonacci Sequence. Let F(n) = F(n-1) + F(n-2). Let F(0) = 4 and F(1) = 10 ( 5 and 9 were chosen randomly). The following chart shows the first 20 terms of this sequence in the left column and the ratio of consecutive terms in the right column.
For a table showing the first 20 terms of this sequence as well as ratios of the consecutive terms of the sequence, click here.
Even more surprising is that this sequence in which the first two terms, 4 and 10, are not even members of the Fibonacci Sequence still have a ratio of consecutive terms that approaches the golden ratio as n approaches infinity.
Summary: This shows us that when the sequence is defined by F(n) = F(n-1) + F(n-2), the limiting value of the ratio of consecutive terms will always approach the golden ratio as n approaches infinity, independent of the first two terms of the sequence. This is reasonable because the definition of the sequence, F(n) = F(n-1) + F(n-2), and the definition of the ratio of consecutive terms of the sequence, make no specifications for the values of the "starter terms", F(0) and F(1). Therefore, the limiting value of the ratio of consecutive terms of a sequence rely not on the starting terms of the sequence but on the definition of the sequence.
For a table showing the ratios of every second term approach its limiting value as the terms in the Fi bonacci Sequence approach inifinity, click here.
As you can see from the table, the limiting value of the ratio of every second term seems to be L = 2.61803399. Why is this?
The ratio of every second term of the Fibonacci Sequence, is defined by:
The limit of this ratio as n approaches infinity is equal to:
We saw earlier that (the limit of as n approaches infinity) = L = 1.61803399.
( the limit of (1) as n approaches infintiy) = 1
Therefore, plugging these values into our definiton of the ratio of every second term of the Fibonacci Sequence, we get:
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