Fibonacci Sequence

by Zack Kroll

For this investigation we examine the Fibonnaci sequence using Excel spreadsheets. The Fibonnaci sequence starts with f(0) = 1, f(1) = 1 followed by the function f(n) = f(n – 1) + f(n – 2). I have always enjoyed looking at sets of numbers and determining how they relate to one another.

Fibonnaci sequence is one of the most talked about sequences in primary and secondary mathematics. When looking at the Fibonnaci sequence it is important to examine the ratios that occur between each of the terms.

Using a spreadsheet we were able to look at the sequence and the adjacent terms. In the two columns we have n and f(n). The goal is to find the ratio of each pair of adjacent terms in the sequence. We decided to look at ratios for the first through the fourth terms to see what changes occur, but also to see if there are similarities.

If we look at the third column (Ratio 1), we see that the ratio shuffles around and eventually settles at 1.61803399 as n increases. The ratios of the second, third, and fourth terms follow a similar pattern. The number eventually settles on a given ratio at approximately the same n value.

After looking at the Fibonnaci sequence we decide to look and see if there were other sequences that have the same limit of the ratio of the successive terms. The first one we investigate is known as the Lucas Sequence. This sequence occurs when f(0) = 1, but f(1) = 3. However, we can still see that the limit of the ratios is almost identical as with the Fibonnaci Sequence.

The final sequence is one that we developed to ensure that the pattern of the ratios was not limited to these two well-known sequences. One key difference between this sequence and the previous two is that all of the values of f(n) are negative. The limit of the ratio still holds.