Real-Life Applications Visual Arts The Fibonacci Sequence “How many pairs of rabbits will be produced in a year, beginning with a single pair if in every month each pair bears a new pair which becomes productive exactly 2 months after birth?” Fibonacci Example #1: Halley's Comet Example #2: Chinese Zodiacs Facts Lucas Series Fibonacci A Closer Look at the Fibonacci Sequence Humans Architecture Nature iris (3 petals) buttercup (5 petals) Delphinium (8 petals) Ragwort (13 petals) Parthenon (Greece) 'Mona Lisa' (Da Vinci) The Perfect Smile United Nations Headquarters (USA) Golden Ratio Golden Rectangle Fibonacci Squares 1.618... (called 'phi') As the nth term of Fibonacci Sequence increases, the ratio of the term and the preceding term gets closer to 1.618 but never reaches the exact value. Fibonacci Spiral Halley's Comet appeared in year 1531, 1607, and 1682, approximately 75 to 76 years apart. If it was not for the gravitational influence of the planets, the Comet would always reappear every 77 years. Question: Given the arithmetic sequence for Halley's Comet since 1531, find tn and the year that will correspond to its 25th appearannce, assumming that there is no gravitational influence from the planets. Solution: a = 1531

d = 77 tn = a + (n-1) d

= (1531) + (n-1) (77)

= 1531 + 77n - 77

= 77n + 1454 Write the general arithmetic sequence formula.

Substitute values for 'a' and 'd.'

Expand.

Collect like terms. Therefore, the formula is: tn = 77n + 1454. Find tn Find the year that corresponds to its 25th appearance. n = 25 tn = 77n + 1454

t25 = 77 (25) + 1454

t25 = 1925 + 1454

t25 = 3379 Write the arithmetic sequence formula.

substitute n = 25 into the equation.

multiply.

add. Therefore, the 25th appearance of Halley's Comet will occur in year 3379. Situation: The Chinese calendar has a 12 year cycle in which one animal is associated with one year, and always appear in that same order. invented the Fibonacci sequence.

introduced the 0 to 9 numeral system in Europe.

presented a problem in his book in 1202 called 'Liber abaci.' Definition: Questions: a) The year 2000 was the year of the dragon. list the next three years that will the year of the dragon.

b) Draw a graph for part a) to show the relationship between the term numbers and their corresponding year of the dragon.

c) Find the formula for the nth term.

d) Will the year 2146 be the year of the dragon? How about the year 2168? The term given when a line is divided into two parts, such that the ratio of the longer section and the smaller section, and the entire length and the longer section are the same. Solution: a) 2012, 2024, and 2036 will be the year ot the dragon. Definition: A spiral formed by connecting the opposite corners of the Fibonacci Squares in the order of the Fibonacci Sequence starting from the 1x1 square. (The pattern repeats every 12 years) c) a = 2000

d = 12 'a' represents the first term in the arithmetic sequence.

'd' represents the common difference Write the general arithmetic sequence formula.

Substitute values for 'a' and 'd.'

Expand.

Collect like terms. The formula for the nth term is: tn = 12n +1988. d) tn = 12n + 1988

(2146) = 12n + 1988

158 = 12n

13.2 = n Write the arithmetic sequence formula for the nth term.

Substitute tn = 2146 into the equation.

Subtract 1988 on both sides of the equation.

Divide by 12 on both sides. Since 'n' is NOT a natural number, 2146 does not exist in the sequence.

Therefore, year 2146 will not the year of the dragon. tn = 12n + 1988

(2168) = 12n + 1988

180 = 12n

15 = n Write the arithmetic sequence formula for the nth term.

Substitute tn = 2168 into the equation.

Subtract 1988 on both sides of the equation.

Divide by 12 on both sides. Since 'n' is a natural number, 2168 exists in the sequence.

Therefore, year 2168 will be the year of the dragon. Definition: A rectangle with the approximate ratio of length to width of 1.618, the golden ratio. A golden rectangle is also considered to be the most visually pleasing. How it Works: Starting off with two squares of length size 1 placed side by side, a square with the successive Fibonacci number is placed on the longer side of the rectangle. The larger the rectangle is, the closer it will become a golden rectangle. Fibonacci first proposed this question: From the problem, this is given:

- Start off with 2 rabbits (1 male, 1 female)

- Each pair will produce a new pair

- Each new pair will become productive after two months

- There are 12 months in a year (x12th term) Fibonacci considers the following as ideal circumstances:

1. The rabbits do not escape therefore Fibonacci will not lose any rabbits

2. The rabbits never die so they can continue to reproduce

3. Every female rabbit will produce only ONE pair of baby rabbits

4. Each pair of newborns consists of a male and female rabbit tn = a + (n-1) d

= 2000 + (n-1) (12)

= 2000 + 12n - 12

= 12n + 1988 'a' represents the first term from the arithmetic sequence.

'd' represents the common difference. 'n' represents the term number. tn = 2146 It represents the value of an nth term in the arithmetic sequence. Significance: Pinecones 8 swirls

CW direction 13 swirls

CCW direction by Edouard Lucas 2, 1, 3, 4, 7, 11, 18... The sequence starts with 2 instead of 1. The golden ratio applies to this sequence. (French mathematician) Binet's Formula by Jacques Binet (French mathematician) Used to find the value of the nth term from the Fibonacci Sequence. Pascal's Triangle by Blaise Pascal (French mathematician) The values in the Sequence can be determined by adding the numbers along a diagonal from top to bottom. b) Formula: t1 = 1, t2 = 1; tn = tn-1 + tn-2. Forearm and Hand tn = 2168 It represents the value of an nth term in the sequeunce.

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