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Algebra-1

You will learn about the Fibonacci Sequence, its history, and its appearance in nature.

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After completing this tutorial, you will be able to complete the following:

- Define the term sequence.
- Identify sequences.
- Describe the Fibonacci Sequence.
- Find specified terms of the Fibonacci Sequence given other necessary terms.

Leonardo of Pisa, now known as Fibonacci, was a medieval Italian mathematician. In 1202, he published his book, Liber Abaci, which contained the following problem:

A pair of rabbits, one male and one female, is placed in a field in January. In February the female becomes pregnant. Her pregnancy lasts for one month, and in March she gives birth to one male and one female rabbit. The female continues to give birth to one new pair of rabbits each month. Each new pair of rabbits matures after one month and begins producing offspring similar to the original pair every month after that. If the rabbits never die, how many pairs of rabbits will there be after months?

This problem is simply referred to as Fibonacci's rabbit problem. The assumptions made within the problem are unrealistic, so the real significance of the problem is the sequence that results, rather than any applicability to population dynamics.

A sequence is an ordered and countable set of numbers. We are familiar with many ordinary sequences; for example, the sequence of natural (or counting) numbers: 1, 2, 3, 4, 5, K, the sequence of positive even numbers: 2, 4, 6, 8, 10, K, and the sequence of positive odd numbers: 1, 3, 5, 7, 9, 11, K. Generally speaking, the terms of a sequence do not need to follow any particular pattern, although the sequences just described do all follow a pattern. The number of pairs of rabbits in each month in the rabbit problem also forms a sequence. The sequence formed by these numbers is called the Fibonacci Sequence.

The Fibonacci Sequence is the sequence given by 1, 1, 2, 3, 5, 8, 13, K. Each term after the first two is equal to the sum of the two previous terms. This interdependence of terms in the Fibonacci Sequence makes it easy to find new terms from old. For instance, the tenth term in the Fibonacci Sequence is 55 and the eleventh term in the Fibonacci Sequence is 89. Since the twelfth term is the sum of the tenth and eleventh terms, we conclude that the twelfth term of the Fibonacci Sequence is 55 +89 =144. Alternatively, if we know that the eighth term of the Fibonacci Sequence is 21, and we recall that the tenth term in the Fibonacci Sequence is 55, then we may also find the ninth term in the Fibonacci Sequence. Since the sum of the eighth and ninth terms equals the tenth term, we may solve the equation 21 +x = 55 for x , where x represents the ninth term in the Fibonacci Sequence, to find that the ninth term of the Fibonacci Sequence is 34.

The Fibonacci Sequence is a pure mathematical object. However, the Fibonacci Sequence is manifested throughout the patterns and shapes of the natural world. One such example of the Fibonacci Sequence in nature is the nautilus shell. We obtain a stunningly accurate model of the nautilus shell from the Fibonacci Sequence as follows: we begin with two squares of side length 1 situated side by side. Next to these, we place a square with side length 2 ( ) to form a rectangle of side lengths 2 and 3. Along the side of length 3, we place a square of side length 3 ( ) to form a rectangle of sides 5 and 3. We continue in this manner to construct what is called the golden rectangle. We construct a curved line through the diagonals of each of the squares to create the golden spiral. This spiral is a model of the spiral found in the nautilus shell. Similar shapes and patterns are found throughout the physical world; the patterns of mathematics are all around us!

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should be familiar with solving linear equations in one variable. |

Course | Algebra-1 |

Type of Tutorial | Skills Application |

Key Vocabulary | Fibonnaci sequence, Fibonnaci spiral, Fibonnaci numbers |