```
Assignment #12 ~ Spreading out the Fibonacci Sequence
The Fibonacci Sequence

By Kevin Mylod
```

Around the late 12th-early 13th century, a man by the name of Leonardo de Pisa, otherwise known as Fibonacci, discovered a specific sequence of numbers that would later be found to have much relevance to many unseemingly related topics. Fibonacci discovered this number sequence while conducting an experiment to find how many rabbits would exist after a certain number of months of mating. Through the results of this experiment, he uncovered a special pattern of numbers relating to the number of rabbits produced. In 1202, he recorded his experiment and results as part of a manuscript entitled Liber Abbaci. The numbers were left alone without much investigation until the late 19th century. In 1878, Frenchman Edwourd Louci publicized the Fibonacci numbers, naming them "The Fibonacci's Sequence." From then on, the sequence has been explored and uncovered many unusual relationships with nature. By working with the Fibonacci Sequence, mathematical patterns have been found in subjects such as botany, biology, art, number theory, astronomy, and music.

```above information was edited from Emily Runge's web site
```

The Fibonacci Sequence

Given the numbers 0 and 1, the Fibonacci sequence is defined as a series of numbers generated by adding the two previous numbers in the sequence.

One can use whatever symbols to define a series of numbers. In this case we will define them as a function, f(n), where n is the numerical position in the sequence. So, for example, f(0) = 0 and f(1) = 1 respectively. Given these first two numbers, the next three numbers in the sequence ensue:

```f(2) = f(0) +f(1) = 0 + 1 = 1
f(3) = f(1) + f(2) = 1 + 1 = 2
f(4) = f(2) + f(3) = 1 + 2 = 3```

We can continue generating an infinite amount of numbers in the sequence using the general function, f(n) = f(n-2) + f(n-1). However the laborious process of generating all of these numbers by hand can be quite time-consuming, especially when f(n) is large. Using a spreadsheet for this sequence will be helpful to generate as many numbers as necessary for whatever Fibonacci investigation you wish to pursue.

So, What is our Investigation?

For the purposes of this write-up, our investigation will discover the relationship between each ratio, f(n)/f(n-1), in the sequence. But first, let's generate the first 35 numbers in the Fibonacci Sequence using the spreadsheet, Microsoft Excel 2000. To show how to set up the spreadsheet we will look at the first five numbers, f(0) through f(4) in the program below.

```                                                                                            Col. A           Col. B             Col. C
f(n)          f(n-2) + f(n-1)   f(n)/f(n-1)```

 ...... A B C 12 f(0) 0 ----- 13 f(1) 1 ----- 14 f(2) 1 1 15 f(3) 2 2 16 f(4) 3 1.5

The above spreadsheet is merely part of the speadsheet created for this investigation. The first column designates the row numbers in the spreadsheet. The second, third, and fourth columns are designated by letters in alphabetical order. The values in column A are nothing more than numerical markers. The values in column B are generated by the formula, f(n-2) + f(n-1). In the Excel program, however, the code for putting in such a formula, for example in cell B14, is "=(B12 + B13)". Once this value is generated, we can simply copy the formula in B14 in as many cells in column B as necessary. For our purposes we will copy the formula down to the 35th number in the sequence, or f(34). Recall that the formula in Column B does not actually begin until cell B14, as values 0 and 1 are given to us. All of the other numbers in the sequence are based upon these two.

The values in column C are generated in a similar fashion as column B, by coding the formula above as "=(B14/B13)" for cell C14. Likewise, we copy this formula for the remaining cells in column C. Notice that there is no ratio in cells C12 or C13 because the formula is not defined for these cells.

Although we cannot see where the ratios in column C are heading, we can see by opening up the full Excel spreadsheet that the ratio does approach what is known as the Golden Ratio, which is (1 + sqrt (5))/2 = 1.618........ . For those of you who have some free time on your hands, click on the numberical value of the Golden Ratio and email me the number of decimal places there are!

Okay, so now we uncover the golden ratio for Fibonacci numbers. What about other numbers? What if we did not start with f(0) = 0 and f(1) = 1? Instead we started with arbitrary numbers f(0) = 11 and f(1) = 29? Will the ratios defined above approach the golden ratio as well? Again, the answer to this question becomes quite clear with the help of the Excel spreadsheet. We can find the results to this investigation in columns E and F. In fact, it does not matter what two numbers we start with, the ratio will eventually work itself out to be the Golden Ratio.

```
To find out more about Leonardo de Pisa, the Fibonacci Sequence, or the Golden Ratio, visit the following web sites

Leonardo Pisano Fibonacci
Who was Fibonacci?
Fibonacci Numbers and the Golden Ratio```
```