Transcript Fibonacci Presentation

Fibonacci
Leonard de Pisa
1175 – 1250 AD
Best remembered for a problem he posed in Liber Abaci dealing
with RABBITS!
The Rabbit Problem
• At 2 months, the rabbits can
reproduce a pair of bunnies.
• How many pairs at k
months?
The Rabbit Problem
Months
Pairs
1
1
The Rabbit Problem
Months
Pairs
2
1
The Rabbit Problem
Months
Pairs
3
2
The Rabbit Problem
Months
Pairs
4
3
The Rabbit Problem
Months
Pairs
5
5
The Rabbit Problem
Months
Pairs
5
8
The next month, all of the gray
rabbits that already existed the
month before, will make more
pairs. The rabbits created this
month will not. So we add,
B(4)+B(5) = B(6)
Recursively defined Fibonacci Sequence
Each rabbit from two months ago reproduces a pair.
So we add that number to the current number of rabbits.
The Fibonacci Sequence
Some Properties of the Fibonacci
Numbers
• No consecutive Fibonacci numbers have a
common factor
F2  F1  0
F3  F2  2  1  1  F1
F4  F3  3  2  1  F2
F5  F4  5  3  2  F3
No consecutive Fibonacci numbers have
a common factor
Suppose d is a common factor for Fn and Fn 1
Then d is a factor of Fn 1  Fn  Fn 1
Then d is a factor of Fn  Fn 1  Fn  2
What can we conclude from this?
Some Properties of the Fibonacci
Numbers
1. Find the sum of the first five terms of the
Fibonacci sequence.
2. Can you find a pattern? Try for six and seven if
you can’t see it right away.
n
F  F  F
i 0
i
1
2
 F3 
 Fn 1  Fn
Some Properties of the Fibonacci
Numbers
F  Fn 1 Fn 1  Fn ( Fn 1  Fn  2 )  Fn 1 Fn 1
2
n
1.
2.
3.
4.
Distribute the right side, then factor out Fn-1
Simplify Fn  Fn 1
Reiterate this process
Use this to show
F  Fn 1 Fn  (1)
2
n
n 1
for n  2