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A Series of Fibs

By: John Vereen

            For my bonus exploration, I will explore the Fibonacci sequence using Excel. What exactly is the Fibonacci sequence? Well, the Fibonacci sequence in a sequence of numbers of the form a_n = a_n-1 + a_n-2. In other words, each successive number of the sequence is the sum of the previous two numbers in the sequence. The Fibonacci sequence starts with the numbers 0 and 1.

            Using Microsoft Excel, we have created four specific columns

Column 1 = Fibonacci sequence

Column 2 = ratio of consecutive terms = 1.618

Column 3 = ratio of every other term = 2.618

Column 4 = ratio of every third term = 4.236

            We notice a recurring theme with the ratio columns of the Fibonacci sequence. Whether it’s the ratio between consecutive numbers, every other number, or every third number, there is always a limit to what the ratio will be. Each ratio column starts out unstable and then levels out to a set ratio.

 

            In the image above, we explored the Lucas sequence. This sequence is similar to the Fibonacci sequence, except we start with 1 and 3 instead of 0 and 1. The columns are as follows:

Column 1 = Lucas sequence

Column 2 = ratio of consecutive terms = 1.618

Column 3 = ratio of every other term = 2.618

Column 4 = ratio of every third term = 4.236

It is fascinating that the columns for each ratio of the consecutive terms, every other term, and every third term are the same for both sequences despite the different initial values! Also, each initial ratio of 1.618 is equivalent to the golden ratio, which is a famous phenomenon of mathematics.