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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The Fibonacci sequence is seen all around us. Learn how the Fibonacci sequence relates to the golden ratio and explore how your body and various items, like seashells and flowers, demonstrate the sequence in the real world.

The **Fibonacci sequence** begins with the numbers 0 and 1. The third number in the sequence is the first two numbers added together (0 + 1 = 1). The fourth number in the sequence is the second and third numbers added together (1 + 1 = 2). Each successive number is the addition of the previous two numbers in the sequence. The sequence ends up looking like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on and so forth.

Looking at it, you can see that each number in the sequence is the addition or sum of the two previous numbers. For example, 34 is the addition of 21 and 13. 144 is the addition of 89 and 55. Try it out yourself and check other numbers in the sequence to see if they follow the rule.

The **golden ratio**, represented by the Greek letter phi, is approximately 1.618. The golden ratio, like pi, is an irrational number that keeps going. The actual value goes like this: 1.618033988764989. . .

You might be wondering how the Fibonacci sequence relates to this number. Let us see.

Let's start by dividing pairs of numbers in the Fibonacci sequence. We will skip zero and start with the pair of ones. 1 / 1 = 1. The next pair is the one and the two. 2 / 1 = 2. In each pair, we divide the larger by the smaller number. Let's keep going and see where it takes us:

Fibonacci pair | Result |
---|---|

2 and 3 | 3 / 2 = 1.5 |

3 and 5 | 5 / 3 = 1.6666. . . |

5 and 8 | 8 / 5 = 1.6 |

8 and 13 | 13 / 8 = 1.625 |

13 and 21 | 21 / 13 = 1.6154. . . |

21 and 34 | 34 / 21 = 1.619. . . |

34 and 55 | 55 / 34 = 1.618. . . |

55 and 89 | 89 / 55 = 1.618. . . |

89 and 144 | 144 / 89 = 1.618. . . |

As the numbers get larger, an interesting thing starts to happen. The result of dividing the pairs of numbers gives you the approximate value of the golden ratio, 1.618. . .

In mathematical terms, the Fibonacci sequence **converges** on the golden ratio. What that means is that, as the Fibonacci sequence grows, when you divide a pair of numbers from the sequence, the result will get closer and closer to the actual value of the golden ratio. Looking at the table, you can see that starting with the pair 34 and 55, the result is accurate to three decimal places. As the pairs get larger, the result will be more accurate to the decimal.

The Fibonacci sequence approximates the golden ratio, which can be found in the natural world. You can see it in your own body, in the way seashells grow, and the number of petals in flowers. Let's take a look at these real-life examples.

Take a look at your own fingers. Notice that each finger has three parts to it. If you measure each section and divide pairs of sections, you will get an approximate value of the golden ratio. If the smallest section of your finger measures one unit, then the section below will measure roughly two units, and the third section will measure about three units. Notice how each measurement corresponds to a Fibonacci number.

Seashells grow in a Fibonacci sequence. If you tile squares with sizes that follow the Fibonacci sequence (1, 1, 2, 3, 5, and so on) and draw a spiral that connects to each outer edge, you will see a seashell forming.

You will also see the Fibonacci numbers in the way flowers grow their petals. An orchid, for example, has several layers of petals and each layer corresponds to a Fibonacci number. Look at the orchid picture, and you will see a petal layer of two, then a petal layer of three, followed by an outer petal layer of 5.

To summarize, the **Fibonacci sequence** begins with 0 and 1, and each successive number is the sum of the two previous numbers. As the Fibonacci sequence grows, if you divide pairs of numbers in the sequence (the larger by the smaller), you will get an approximate value of the golden ratio, which is roughly 1.618.

When you are finished, you should be able to:

- Describe and identify the Fibonacci sequence
- Explain the relationship between the Fibonacci sequence and the golden ratio
- Recall some examples of the Fibonacci sequence in nature

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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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