Excel SpreadSheet Exploration of the Fibonacci Sequence

By

Nathan Wisdom

Problem: Using Excel, generate the Fibonacci Sequence in the first column such that f(1)=1, f(2) =1, f(n) = f(n-1) + f(n-2) for integer n greater than or equal to 3.

Fibonacci explored the problem:

Fibonacci explored the problem: A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

To explore this problem using Excel Spreadsheet, we enter 1 in cell A1 and 1 in cell A2. In cell A3 we enter the formula (A2 +A1). We now “drag” A3 to fill Column A. This will generate the Fibonacci sequence in column A. See the SpreadSheet.

We now construct the ratio of each pair of adjacent terms in the Fibonacci sequence.

We Now look at the ratio:f(n+1)/f(n), f(n+2)/f(n), f(n+3)/f(n), f(n+4)/f(n) etc.

Therefore , in cell B1 we enter the formula (A2/A1), in C1 enter A3/A1, in D1 enter A4/A1, etc.

Note that as the integer n increases, the sequence converge to a constant, and the ratio of every second term converges to , ( approximately by the 16th term).

Finally, the ratio of every second term converges to 1. Now use the spread sheet provided to explore other sequences. For instance, if f(1) =1 and f(2)=3, then the sequence is a Lucas Sequence. Note that the ratio of every second term for these sequences also converges to 1.

Please Explore the spread sheet provided. There are several pages named FIBO1, FIBO2, etc. Each page is connected to a previous page so you can make changes in one spreadsheet and the sequence will change on the other sheets. Also please look at Excel the formulae.