 Fibonacci Sequences

By Pei-Chun Shih

In the year 1202, the Italian mathematician Leonardo de Pisa, best known as Fibonacci, presented his famous findings about how fast rabbits could breed in ideal circumstances.

Here are the assumptions:

1. Suppose a newly-born pair of rabbits, one male, one female, are

allowed to breed in a controlled environment.

2. Rabbits are able to mate at the age of one month.

3. At the end of its second month a female can produce another pair of

rabbits.

4. The rabbits never die and the female always produces one new pair

(one male, one female) every month from the second month on.

How many pairs will there be in one year? There will be 233 pairs of rabbits during the twelfth month.

The rabbits’ problem generated the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …, which is known today as the Fibonacci Numbers and the sequence is called the Fibonacci Sequence. Every numbers in the Fibonacci sequence (after the first two) is the sum of the two preceding numbers. Therefore, the Fibonacci numbers can be defined as the linear recurrence equation: Fn = Fn-1 + Fn-2 with F0 = 0 and F1 = F2 = 1.

Hence, F3 = F2 + F1 = 1 + 1 = 2

F4 = F3 + F2 = 2 + 1 = 3

……

F13 = F12 + F11 = 144 + 89 = 233, and so on.

After understanding the origin of the Fibonacci numbers and sequence, I am going to use the spreadsheet to do some explorations.

First, I generate a Fibonacci sequence in the second column of the spreadsheet using F(0) = 1 and F(1) = 1. Second, in the third column I construct the ratio of each pair of adjacent terms in the Fibonacci sequence, that is Fn/Fn-1. In the fourth column, I construct the ratio of every second term in the Fibonacci sequence. Likewise, the fifth column is the ratio of every third term in the Fibonacci sequence, and the sixth column is the ratio of every fourth term in the Fibonacci sequence, and so forth. As we can see from the third column, as the numbers get larger, the ratio seems to approach a specific number. What is this specific number that the ratios are approaching?

Let’s sketch a diagram to help us visualized the pattern. Let n be the x-coordinate and (Fn / Fn-1) be the y-coordinate. The diagram above also shows that the ratio of two neighboring Fibonacci numbers soon settled down to a particular value near 1.61803 as n increases. This specific number, 1.618033989, is known as the Golden Ratio.

The numbers in the fourth column, which are the ratios of every second term, also approach to a specific number as n getting large. This specific number is the second power of the golden ratio, that is (1.618033989)2 = 2.618033989. Similarly, the numbers of the fifth column come close to the third power and the sixth column to the fourth power of the golden ratio as n increases.

Another interesting discovery can be found from the spreadsheet above that the first two cells of each column also form the Fibonacci numbers.

Although the Fibonacci numbers had existed since the year 1202, they didn’t get too much attention until a French mathematician, Edouard Lucas (1842-1891), studied them in the second half of the nineteenth century. He was curious about the start of the Fibonacci sequence and wondering what would happen if the sequence had begun with 1 and 3. So he started his number with 1 and 3, rather than 1 and 1 and followed the same additive rule as the Fibonacci numbers. He found that the Lucas numbers, which are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, …, interconnect with the Fibonacci numbers. Here I am going to use the spreadsheet to explore their relationship. In the spreadsheet above, I generated the Lucas numbers using the same technique I did in generating the Fibonacci numbers earlier. From the third column above, it also shows that the ratio of two neighboring Lucas numbers soon settled down to the golden ratio, 1.618033989, as n increases.

Like the ratios of the Fibonacci numbers, the Lucas numbers in the fourth column, which are the ratios of every second term, also approach to the second power of the golden ratio, that is (1.618033989)2 = 2.618033989. Similarly, the numbers of the fifth column come close to the third power and the sixth column to the fourth power of the golden ratio as n increases. Likewise, the first cell of each column also forms the Lucas numbers.

Now we know that both the Fibonacci and the Lucas sequences approach the golden ratio. Do they have direct connection with each other? Let’s use the spreadsheet to try to relate these two sequences directly.

After several attempts, I realized there is a direct connection between these two sequences. The nth Lucas number (n ³ 0) is equal to the sum of the (n-1)st Fibonacci number and the (n+1)st Fibonacci number. That is: Ln = Fn-1 + Fn+1. In the diagram below, you can observe this relationship from the first four columns where the sequences placed next to each other. When is a Fibonacci number equal to a Lucas number?

From the spreadsheet above, it seems that they only equal at the very first beginning, that is F1 = L1 = 1, F2 = L1 =1, and F4 = L2 = 3. They have never got any other chance to be equal as n increases.

I will close this write-up with the following question: can we use GSP to generate the Fibonacci sequence? The answer is yes, but how?

First, we have to define the starting value of the sequence and the difference. GSP uses “seed” as the command to generate a sequence. In order to differentiate the sequence, GSP uses square brackets [] to indicate the number within will be a subscript. Thus, “seed ” will define the first term of the arithmetic sequence. But where should we place the command?

Choose Calculate under the menu of Measurement. Select Value then New Parameter. In the box for Name, type “seed ”. In the box for Value, type 1. Also define seed  = 1 in the same way. These two steps define the first and the second terms of the Fibonacci numbers. Use Calculate to calculate “seed1 + seed2” and “seed2 + (seed1 + seed2)” which is the third and the fourth terms respectively.

Second, we need to know how the Transform menu’s Iterate command functions. According to the help menu of the GSP, Iterate command enables you visualize the orbit of one or more objects over some number of repetitions of a construction. Therefore, by selecting seed1 and seed2, then choose Iterate from the Transform menu and do things like the diagram below, we can generate the Fibonacci sequence we want. The idea here is to iterate with seed1 = 1 mapped to seed2 =1 and seed2 =1 mapped to (seed1 + seed2) = 2. You can choose the number of iterations at the Display dialog box.

Voila! Here is the Fibonacci numbers generating by GSP. The last, I plot the Fibonacci numbers to help visualize the dramatic change of them. The plot below shows how fast and steeply the values grow. 