Initially:

1
Initially
1 pair of juvenile rabbits
2
After 1 month
The pair of rabbits are now mature
3
After 2 months
The original pair of rabbits are (still) mature,
and there is now a new pair of juvenile offspring.
4
After 3.
5

And 4...
6
And 5...
7
And 6...
8
And 7...
9
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits
10
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
11
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1
Any rabbits that were around initially are mature
after 1 month.
12
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2
Any rabbits that were around after 1 month are
mature after 2 months. Any rabbits that were
mature after 1 month give birth to offspring in
the 2nd month.
13
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2 4 1 2 3
Any rabbits that were around after 2 months are
mature after 3 months. Any rabbits that were
mature after 2 month give birth to offspring in
the 3rd month.
14
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2 4 1 2 3 5 2 3 5
15
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2 4 1 2 3 5 2 3 5
6 3 5 8
16
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2 4 1 2 3 5 2 3 5
6 3 5 8 7 5 8 13
17
During this there are this many and this
many pairs total number month pairs
of young rabbits of mature rabbits 1 1 1
2 1 1 3 1 1 2 4 1 2 3 5 2 3 5
6 3 5 8 7 5 8 13 8 8 13 21
18
General Pattern
In any given month, the total number of rabbits
number of mature rabbits number of juvenile
rabbits.
19
General Pattern
In any given month, the total number of rabbits
number of mature rabbits number of juvenile
rabbits. But the number of mature rabbits the
total number of rabbits from the previous month,
and the number of juvenile rabbits the number
of mature rabbits from the previous month, which
is the same as the total number of rabbits from 2
months prior.
20
General Pattern
In any given month, the total number of rabbits
number of mature rabbits number of juvenile
rabbits. But the number of mature rabbits the
total number of rabbits from the previous month,
and the number of juvenile rabbits the number
of mature rabbits from the previous month, which
is the same as the total number of rabbits from 2
months prior. Conclusion The number of rabbits
in any month can be found by adding the number of
rabbits one month earlier to the number of
rabbits 2 months earlier.
21
General Pattern
Let fn represent the total number of rabbits
after n months. Then, we have the formula fn
fn-1 fn-2 number of rabbits after
number after number after n months n-1
months n-2 months
22
After this many months there are this many
pairs of rabbits 1 1 2 1 3 2 4 3 5
5 6 8 7 13 8 21 General
formula fn fn-1 fn-2
23
After this many months there are this many
pairs of rabbits 1 1 2 1 3 2 4 3 5
5 6 8 7 13 8 21 General
formula fn fn-1 fn-2 with initial
conditions f1 f2 1.
24
The Fibonacci sequence

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

25
The Fibonacci sequence

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
• What is the next Fibonacci number?

26
The Fibonacci sequence

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
• 345589, so 89 comes next.

27

• A sequence is recursively defined if, as in this
  example, the rule for finding the next number is
  based on the numbers already found.

28

• Another example of a recursively defined sequence
  might be 10, 11, 12, 13
• What comes next? What is the recursive rule?

29

• Recursively defined sequences are often easy to
  work with, but they do suffer from one serious
  drawback

30

• Recursively defined sequences are often easy to
  work with, but they do suffer from one serious
  drawback
• What is the 30th Fibonacci number?

31

• An alternative way to define a sequence is to
  explicitly define the terms give an explicit
  formula, or algebraic rule, which tells you how
  to compute the nth number directly.

32

• Consider the sequence defined by the rule
• ann2.
• What is the 1st term, a1?
• What is the 2nd term, a2?
• What is the 30th term?

33

• Lets return for a moment to the sequence
• 10, 11, 12, 13,
• What is an explicit formula for this sequence?

34

• 10, 11, 12, 13,
• Since the first number is 10, we know we want
  a110

35

• 10, 11, 12, 13,
• Since the first number is 10, we know we want
  a110
• Also, we want our formula to cause an to increase
  by 1 every time n is increased by 1.

36

• 10, 11, 12, 13,
• an9n meets both these requirements, so this is
  our explicit formula for this sequence.

37

• As with the last example, there is an explicit
  formula for the Fibonacci numbers, too. It just
  is a little messier

38
The Fibonacci sequence, explicit formula

• This is known as Binets formula

39

• The number is important enough that it is
  given a symbol, F, the Greek letter Phi.
• A large portion of this chapter will deal with
  properties of F.