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Sequence

Sequence. Chapter 6 section 6. Definition. What is a sequence? Sequence is a list of numbers that follows some rule or pattern. Types of Sequences. 1. Arithmetic Sequence- is a sequence in which each term after the first term differs from the preceding term by a fixed constant. Examples;

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Sequence

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  1. Sequence Chapter 6 section 6

  2. Definition What is a sequence? Sequence is a list of numbers that follows some rule or pattern.

  3. Types of Sequences • 1. Arithmetic Sequence- is a sequence in which each term after the first term differs from the preceding term by a fixed constant. • Examples; • 3, 7, 11, 15, … • -6, -2, 2, 6, …

  4. Geometric sequence- is a sequence in which each term, after the first term is a nonzero constant multiple of the preceding term. The constant by which we are multiplying is called the common ratio. • Examples; • 3, -6, 12, -24, 48,… • 2, 1, ½, ¼, 1/16, …

  5. Identify the sequence as either arithmetic or geometric. List the next two terms.

  6. Problems 11, 7, 3, -1, … 1, ½, ¼, 1/8, … 1, -1, 1, -1,… 10, 5, 0 , -5, …

  7. Solutions Arithmetic; -5, -9 Geometric; 1/16, 1/32 Geometric; 1, -1 Arithmetic; -10, -15

  8. Famous Sequences • Fibonacci • Fibonacci was also known as Leonardo of Pisa (1202). • Wrote a book, Liber Abaci, in which he introduced the Hindu-Arabic numeration system to Europe. • Introduced the most famous sequence in the history of mathematics; The Fibonacci Sequence.

  9. What is the Fibonacci Sequence? It is a sequence that was well known in ancient India (200BC), before Fibonacci introduced it to the west. He considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly-born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

  10. At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

  11. Solution In one year you will have 233 pairs.

  12. Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ,……….

  13. Where does it occur? • This sequence occurs in nature in many different ways. • Examples; • The seeds of daisies and sunflowers, are arranged in spirals going in two different directions. • Hexagonal “Bumps” on the skin of pineapple.

  14. Golden Ratio • What is the Golden Ratio? • How is it related to Fibonacci sequence?

  15. I have answers Known as φ (phi). By dividing each term by the term preceding it. (See next slide)

  16. Squares

  17. Why is this useful? Architects still use this today United Nations building. Leonardo Di Vinci may have used it in his paintings.

  18. Homework

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