By Courtney Cody

In this assignment, I will explore the Fibonacci sequence using spreadsheets.  I will begin by generating a Fibonacci sequence in the first column of a spreadsheet.  Then I will explore the ratios of different terms of the sequence.  Lastly, I will investigate the limit of the sequence when the starting value is changed.

LetŐs begin our investigation on the Fibonacci sequence by defining what exactly such a sequence is.  In mathematics, the following recurrence relation defines the Fibonacci sequence:

Using our definition of the Fibonacci sequence, we know that the first two terms are 1 and that each subsequent term is the sum of the previous two terms.  To generate a Fibonacci sequence in a spreadsheet, we will let the first two entries of the first column be 1.  In all remaining cells of the first column, we will write a formula to create the Fibonacci sequence, which is , where n is the value of each cell.  Hence, the first column of our spreadsheet is the first 35 terms of the Fibonacci sequence and appears as follows:

Suppose we wanted to construct the ratio of each pair of adjacent terms in the Fibonacci sequence.  In order to find these ratios, we want to use the formula .  We will construct these ratios in the second column of the spreadsheet.

It appears as though as n increases, the ratio of each pair of adjacent terms is converging to a particular number.  We have created a chart of the ratios as n increases.  With each term of the series we observe a limiting characteristic.  The value the sequence of ratios is limiting to, 1.61803399É, is a very special number in mathematics. ItŐs called the

Which is defined by .

Now we want to find the ratio of every second term.  Then we will calculate the ratio  for each term in the Fibonacci sequence and place these values in the third column of the spreadsheet.

Once again, we see that the ratios converge to a particular value.  If we compare the second and third columns of the spreadsheet, we notice that for each entry in the third column, it is always exactly one more than the value at each corresponding entry in the second column.  So it we were to calculate the ratio of every third term, should we expect that the value the ratios will converge to be exactly two more than the value of each entry in the second column?  LetŐs investigate!  What if we found the ratio of every third term, fourth term, etc.  Then we would be looking at the ratios , , ,É.  If we calculate these ratios in the next several columns, we have the following:

If we study each column, it is obvious that the ratios in each converge to some value.  Is there anything special about these values?   It is clear that the limit of each ratio does not increase by one each time.  Instead, the numbers appear to grow exponentially.  This observation is confirmed by creating a bar graph to show how the ratios increase.

Hence, the limiting ratios are powers of the Golden Ratio, .  Since the limiting values increase exponentially, we can write a formula to calculate the limiting ratio of various pairs of terms, which is

What if when we defined our sequence we let f(0) and f(1) be some arbitrary integers other than 1?  For instance, let f(0)=2 and f(1)=5.  Keeping all formulas in the spreadsheet the same, we obtain the following:

Like the Fibonacci sequence, we see this new sequence has the same limit of the ratio of successive terms, which are the powers of the Golden Ratio.  LetŐs try another set of initial values, sat f(0)=2 and f(1)=1, which define the Lucas Sequence.  Hence, we obtain the following spreadsheet representing the Lucas Sequence and the ratios of different terms in the sequence:

By studying the spreadsheet, we see that the values in columns B through H for the Lucas sequence have the same limit of the ratio of successive terms as the Fibonacci sequence, which are the powers of the Golden Ratio.  Thus, we can infer from the spreadsheets that no matter what f(0) and f(1) are, if a sequence is defined by , then the limit of the ratio of successive terms will always be the powers of the Golden Ratio.

Is there a relationship between the Fibonacci sequence and the Lucas sequence?  LetŐs place the two sequences side by side and compare the differences of these two sequences.

If we study the two purple shaded regions, we see they are the same!  Hence, we have the formula , or , which expresses the Lucas sequence in terms of the Fibonacci sequence.

There exists another interesting relationship between the Fibonacci sequence and the Lucas sequence.  What if we found the sequence f(2n) and then found the ratio of f(2n) to f(n)?

We notice in the spreadsheet above that the aforementioned ratio is identical to the Lucas sequence!  Hence, we have another formula , which expresses the Lucas sequence in terms of the Fibonacci sequence.

Thus, a spreadsheet is a useful tool for expressing and comparing number sequences, such as the Fibonacci and Lucas sequences.