7.4 Exploring recursive sequences fibonacci

Homefun  pg 441-443 A-E (omit Lucas Sequence on C-E) pg 443 #1-3 Plus: Determine a recursive formula for the sequence 39, 83, 171, 347, 699, …

REVIEW Sequence: an ordered list of numbers Term: a number in a sequence (the first term is referred to as t1, the second term as t2, etc…) example 3, 7, 11, 15, … t1 = 3 t2 = 7 t3 = 11 t4 = 15

REVIEW - Recursive SEQUENCE a sequence for which one or more terms are given each successive term is determined by performing a calculation using the previous term(s) example t1 = 2 describes 2, 6, 18, 54, … tn =3 tn-1 t2 =3t1 =3(2) = 6 n>1 , n  N t3 =3t2 =3(6) = 18

REVIEW - General term a formula that expresses each term of a sequence as a function of its position labelled tn example tn = 2n describes 2, 4, 6, 8, 10

Fibonacci In his book Liber Abaci (The Book of Calculation), Italian mathematician Leonardo Pisano (117021250), nicknamed Fibonacci, described a situation like this: A man put a pair of newborn rabbits (one male and one female) in an area surrounded on all sides by a wall. When the rabbits are in their second month of life, they produce a new pair of rabbits every month (one male and one female), which eventually mate. If the cycle continues, how many pairs of rabbits are there every month?

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, … What are the next five terms? Can you come up with a recursive formula that describes how the terms are generated?

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, … This sequence models the number of petals on many kinds of flowers, spirals on a pineapple, spirals of seeds on a sunflower head, spirals on a pinecone and other naturally occurring phenomena.

Fun & Beautiful Fibonacci numbers 1 1 2 3 5 8 13 21 34 55 89 144 233 377

Fun & Beautiful Fibonacci numbers 1 1 2 3 5 8 13 21 34 55 89 144 233 377

Fun & Beautiful Fibonacci numbers 1 1 2 3 5 8 13 21 34 55 89 144 233 377

Fun & Beautiful 1 1 2 3 5 8 13 21 34 55 89 144 233 377

FIBONNACI SPIRAL

FIBONNACI SPIRAL

FIBONNACI SPIRAL

FIBONNACI SPIRAL

THE GOLDEN RATIO 1.61818

FIBONNACI SEQUENCE & GOLDEN RATIO

FIBONACCI & GOLDEN RATION https://www.youtube.com/watch?v=Wcq5x8rSMXo https://www.youtube.com/watch?v=P0tLbl5LrJ8 https://www.youtube.com/watch?v=nt2OlMAJj6o https://www.youtube.com/watch?v=_GkxCIW46to https://www.youtube.com/watch?v=9CiS3SU4lk0 https://www.youtube.com/watch?v=Zs7xLKCceWg

Example 1 The French mathematician Edouard Lucas studied a sequence that followed the same pattern as the Fibonacci sequence but started with the terms t1 =1 , t2 = 3 Determine the first 10 terms of the Lucas sequence.

LUCAS sequence 1, 3, 4, 7, 11, 18, 29, …

Homefun  pg 441-443 A-E (omit Lucas Sequence on C-E) pg 443 #1-3 Plus: Determine a recursive formula for the sequence 39, 83, 171, 347, 699, …