Assignment 12:  Fibonacci Sequence

By:

Jonathan Sabo

Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1,

f(n) = f(n-1) + f(n-2)

a. 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.

b. 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.

We will find the Fibonacci Ratio by adding consecutive terms.  We will plug  0 and 1 into the first 2 rows of the spreadsheet and create a formula to calculate each of the following terms of the Fibonacci Sequence.  After finding the Fibonacci Sequence we will find the ratio of adjacent terms of the sequence.  Notice that the ratio moves up and down until it settles on the Golden Ratio.  Observe,

Next we will find the second ratio where we will find the ratio between every other term.  After finding this ratio I notice that the ratio between every other term bounces around just like the first case.  This ratio eventually settles on what appears to be the Golden Ration + 1.  Observe,

Now we will find the 3rd ratio.  In order to do this we will find the ratio between every thrid term.  Similar to the previous ratios the we notice that the ratio initially bounces around then settles on a constant ratio in the end.  The 3rd Ratio seems to eventually settle on what appears to be the Adjacent Ration + The 2nd Ratio.  Observe,

After finding additional ratios we notice that each ratio appears to settle on the sum of the previous 2 ratios.  Observe,

Now we will Explore sequences where f(0) and f(1) are some arbitrary integers other than 1.

For the first sequence I chose f(0) = 17 and f(1) = 23.  For the second sequence I chose the Lucas sequence where f(0) = 1 and f(1) = 3.  Observe,

Notice that  each of the ratios are exactly the same as before when we had the fibonacchi sequence.  Lets prove that the fibonacci sequence always converges on the Golden Ratio.
We want to show