University:
Davenport University
Course:
MATH 120 | College Mathematics
Academic year:

2022
Views:

451

Pages:

3
Author:

trescandogss

Fibonacci Sequence & Golden Ratio Fibonacci was an Italian mathematician from Pisa, who invented the following sequence in the year 1202: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ... After the first two terms, each term is formed by adding the two previous terms. Fibonacci used 1 and 1 as his starting numbers, but any two numbers can be used So here, 2=1+1 3=2+1 5=3+2 8=5+3 13 = 8 + 5 etc.... Example Work out the missing term in this Fibonacci sequence: 2, 2, 4, ?, 10, 16 ?... Answer Look at the first two terms, which add to give 4, the third term. The missing term is therefore 2 + 4 = 6 Check that the next terms works too 4 + 6 = 10 and 6 + 10 = 16. Notation for terms in Fibonacci sequence. F0 = 0 F1 = 1 F2 = F0 + F1 = 0 + 1 = 1 F3 = F1 + F2 = 1 + 1 = 2 F4 = F2 + F3 =1 + 2 =3 F5 = F3 + F4 = 2 + 3 = 5 F6 = F4 + F5 = 3 + 5 = 8 F7 = F5 + F6 = 5 + 8 = 13 Question 1) How to find the third term out of 3 terms in Fibonacci sequence, if we know the last two. Answer. Just add the last two. Example: If F12 = 144 and F13 = 233 , find F15 = ? First we need to find F14 = F12 + F13 = 144 + 233 = 377 , then F15 = F13 + F14 = 233 + 377 = 610 . Practice: F20 = 6765 and F21 = 10946 , find F24 = ? Answer is 46368 Question 2) How to find the first term out of 3 terms in Fibonacci sequence, if we know the last two terms. Answer. Just subtract the smaller from the larger. Example: If F12 = 144 and F13 = 233 , find F11 = ? F11 = F13 − F12 = 233 − 144 = 89 , in the same fashion, we can find F10 = F12 − F11 = 144 − 89 = 55 then F15 = F13 + F14 = 233 + 377 = 610 . Practice: F20 = 6765 and F21 = 10946 , find F19 = ? and F18 = ? Answers: F19 = 4181 and F18 = 4181 Golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of two consecutive terms in Fibonacci sequence. Notation for Golden ratio is the Greek alphabet ϕ (pronounced fee). How to approximate ϕ by dividing each term by its previous one. ϕ Example: = ϕ Find = F7 = F6 F10 55 F27 196418 = = 1.617647 or = ϕ = = 1.618033 F9 34 F26 121393 ? = ? Application of Golden ratio: The Golden Ratio is a common mathematical ratio found in nature, which can be used to create pleasing, organic-looking compositions in your design projects or artwork • • • • • • • • Some examples: Flower petals. number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55. ... Faces. Faces, both human and nonhuman, abound with examples of the Golden Ratio. ... Body parts. ... Seed heads. ... 5. Fruits, Vegetables and Trees. ... Shells. ... Spiral Galaxies. ... Hurricanes.