1. 7.4 Exploring recursive sequences fibonacci

2. 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, …

3. 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

4. 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 = describes 2, 6, 18, 54, …
tn =3 tn t2 =3t1 =3(2) = 6
n>1 , n  N t3 =3t2 =3(6) = 18

5. 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

6. Fibonacci
In his book Liber Abaci (The Book of Calculation), Italian mathematician Leonardo Pisano ( ), 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?

7. 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?

8. 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.

9. Fun & Beautiful Fibonacci numbers

10. Fun & Beautiful Fibonacci numbers

11. Fun & Beautiful Fibonacci numbers

12. Fun & Beautiful

13. FIBONNACI SPIRAL

14. FIBONNACI SPIRAL

15. FIBONNACI SPIRAL

16. FIBONNACI SPIRAL

17. THE GOLDEN RATIO

18. FIBONNACI SEQUENCE & GOLDEN RATIO

19. FIBONACCI & GOLDEN RATION

20. 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.

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

24. 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, …