Fibonacci Sequence

College Faculty à Emilio Aguinaldo College
21 Nov 2020
1 sur 20

Fibonacci Sequence

• 2. Objectives: At the end of the lesson, you should be able to perform the following tasks: 1. Define fibonacci sequence and give its significance. 2. Find the next terms of the given fibonacci sequence. 3. Investigate the relationship between the Fibonacci sequence with the golden ratio.
• 3. Who Was Fibonacci? • European Mathematician 1175- 1250 • Real name Leonardo of Pisa. • Author of Liber Abbaci or Book of the Abacus or Book of Calculation
• 4. What Is the Fibonacci Sequence and Why Is It Significant? • Generalized sequence of first two positive integers and the next number is the sum of the previous two, i.e. 1,1,2,3,5,8,13,21 ,…
• 5. What Is the Fibonacci Sequence and Why Is It Significant? • Shows up unexpectedly in architecture, science and nature (sunflowers & pineapples).
• 6. What Is the Fibonacci Sequence and Why Is It Significant? • Has useful applications with computer programming, sorting of data, generation of random numbers, etc.
• 9. Fibonacci Sequence • The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... • The next number is found by adding up the two numbers before it. • The 2 is found by adding the two numbers before it (1+1) • The 3 is found by adding the two numbers before it (1+2), • And the 5 is (2+3), and so on!
• 10. Makes a Spiral When we make squares with those widths, we get a nice spiral:
• 11. The Rule xn = xn-1 + xn-2 where: xn is term number "n" xn-1 is the previous term (n-1) xn-2 is the term before that (n-2)
• 12. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ... Example: the 8th term is the 7th term plus the 6th term: x8 = x7 + x6 First, the terms are numbered from 0 onwards like this: So term number 6 is called x6 (which equals 8).
• 13. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...  Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)  Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)  Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)
• 14. Golden Ratio • In mathematics, two quantities are in the Golden ratio if their ratio is the same of their sum to the larger of the two quantities. • “De Devina Proportione” by Luca Paciolli
• 15. Golden Ratio • Is the relationship between numbers on the Fibonacci sequence where plotting the relationships between on scales results in a spiral shape.
• 19. Seatwork If the first three fibonacci numbers are given as 1,1,2,…, then what is the least value of n for which Xn > 500? Show your solutions.