Assignment 12

This activity focuses on using spreadsheets, such as Microsoft Excel, to explore mathematical concepts such as Fibonacci sequences.

To use Excel to generate a Fibonacci sequence start with the number 1 in block A2, and the number 1 in block A3.

To create a formula that will generate a list of n Fibonacci numbers, click on cell A4 and enter [=A2+A3] in the formula bar.  Then highlight cell A4, copy, and paste into n-2 additional blocks in column A.

Click on cell A30 to check the formula.  In the formula bar, [=A28+A29] appears, so our formula works.

Next let's look at the ratio between each pair of adjacent Fibonacci numbers.  In cell B3, type [=A3/A2].  Copy and paste this formula into the rest of the cells in row B, just like we did for the Fibonacci.  Again, check the formula to be sure it works.

Notice what happens to this ratio as the pairs of Fibonacci numbers increase in value.  As the value of n increases, the ratio oscillates and settles out at 1.618033989.  We can summarize this as follows: As n approaches infinity, the limit of the ratio of adjacent Fibonacci numbers approaches 1.618033989.

Now let's consider a different case.  Choose two arbitrary numbers for f(0) and f(1) and observes what happens to the same ratio.  Let f(0) = 4, and f(1) = 7.

The values in the first column are very different from the sequence of Fibonacci numbers we just created.  However, notice the similarities in the ratios.  The limit of both sets of ratios approach 1.618033989 as n approaches infinity.  In fact, sequence two reaches this limit three numbers earlier than the Fibonacci sequence.

Finally, let’s consider one more case.  Let f(0) = 1 and f(1) = 3.  This is referred to as the Lucas Sequence.

Again, notice the similarities in the ratios.  The limit of all three sets of ratios approach 1.618033989 as n approaches infinity.