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The Fibonacci Numbers
A sequence of numbers (actually, positive integers) that occurmany times in
nature,
art,
music,
geometry,
mathematics,
economics,
and more....
The Fibonacci Numbers
The numbers are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Each Fibonacci number is the sum of the previous two Fibonaccinumbers!
Definition
The nth Fibonacci number is written as Fn.
F3 = 2, F5 = 5, F10 = 55
We have a recursive formula to find each Fibonacci number:
Fn = Fn−1 + Fn−2
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
What makes these numbers so special?
They occur many times in nature,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
What makes these numbers so special?
They occur many times in nature,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
What makes these numbers so special?
They occur many times in nature, art,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
What makes these numbers so special?
They occur many times in nature, art, architecture, . . .
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
white Calla Lily
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
euphorbia
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
trillium
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
columbine
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
bloodroot
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
black-eyed susan
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
shasta daisy
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
field daisies
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
back of passion flower
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
How many petals are on each of these flowers?
front of passion flower
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Let’s look at these Fibonacci numbers in nature.
There are exceptions....
fuschia
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
What do we see?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
The stamen form spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
How many are there?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
How many are there?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
Not just in daisies. In Bellis perennis:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Looking more closely at the blossom in the center of the flower:
Not just in daisies. In Bellis perennis:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
These numbers occur in other plants as well:
The seed-bearing leaves of a simple pinecone:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
These numbers occur in other plants as well:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
These numbers occur in other plants as well:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
These numbers occur in other plants as well:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
These numbers occur in other plants as well:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like cauliflower:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like cauliflower:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like cauliflower:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like romanesque:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like romanesque:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like romanesque:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like pineapples:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like pineapples:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like pineapples:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like pineapples:
Find and count the number of spirals:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
In fruits and vegetables, like pineapples:
Find and count the number of spirals:
The Fibonacci Numbers
Let’s examine some interesting properties of these numbers.
The numbers are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Each Fibonacci number is the sum of the previous two Fibonaccinumbers!
Let n any positive integer. If Fn is what we use to describe the nth
Fibonacci number, then
Fn = Fn−1 + Fn−2
The Fibonacci Numbers
For example, let’s look at the sum of the first several of thesenumbers.
What is F1 + F2 + F3 + F4 · · · + Fn = ???
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Let’s look at the first few:
F1 + F2 = 1 + 1 = 2
F1 + F2 + F3 = 1 + 1 + 2 = 4
F1 + F2 + F3 + F4 = 1 + 1 + 2 + 3 = 7
F1 + F2 + F3 + F4 + F5 = 1 + 1 + 2 + 3 + 5 = 12
F1 + F2 + F3 + F4 + F5 + F6 = 1 + 1 + 2 + 3 + 5 + 8 = 20
See a pattern?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
It looks like maybe the sum of the first few Fibonacci numbers isone less another Fibonacci number!
But which Fibonacci number?
F1 + F2 + F3 + F4 = 1 + 1 + 2 + 3 = 7 = F6 − 1
The sum of the first 4 Fibonacci numbers is one less the 6th
Fibonacci number!
F1 + F2 + F3 + F4 + F5 + F6 = 1 + 1 + 2 + 3 + 5 + 8 = 20 = F8 − 1
The sum of the first 6 Fibonacci numbers is one less the 8th
Fibonacci number!
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
It looks like the sum of the first n Fibonacci numbers is one lessthe (n + 2)nd Fibonacci number.
That is,F1 + F2 + F3 + F4 + · · · + Fn = Fn+2 − 1
Let’s check the formula, for n = 7:
F1+F2+F3+F4+F5+F6+F7 = 1+1+2+3+5+8+13 = 33 = F9−1
It’s got hope to be true!
Let’s try to prove it!
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Conjecture
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F2 + · · · + Fn = Fn+2 − 1
Let’s prove this (and then we’ll call it a Theorem.)
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
We know F3 = F1 + F2.
So, rewriting this a little:
F1 = F3 − F2
Also: we know F4 = F2 + F3.
So:F2 = F4 − F3
In general:
Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3 − F2
F2 = F4 − F3
F3 = F5 − F4
F4 = F6 − F5
......
Fn−1 = Fn+1 − Fn
Fn = Fn+2 − Fn+1
Adding up all the terms on the left sides will give us somethingequal to the sum of the terms on the right sides.
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3 − F2
F2 = F4 − F3
F3 = F5 − F4
F4 = F6 − F5
......
Fn−1 = Fn+1 − Fn
+ Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4 − F3///
F3 = F5 − F4
F4 = F6 − F5
......
Fn−1 = Fn+1 − Fn
+ Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5 − F4///
F4 = F6 − F5
......
Fn−1 = Fn+1 − Fn
+ Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5///− F4///
F4 = F6 − F5///
......
Fn−1 = Fn+1 − Fn
+ Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5///− F4///
F4 = F6///− F5///
......
Fn−1 = Fn+1 − Fn///
+ Fn = Fn+2 − Fn+1
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5///− F4///
F4 = F6///− F5///
......
Fn−1 = Fn+1////// − Fn///
+ Fn = Fn+2 − Fn+1//////
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5///− F4///
F4 = F6///− F5///
......
Fn−1 = Fn+1////// − Fn///
+ Fn = Fn+2 − Fn+1//////
F1 + F2 + F3 + · · · + Fn = Fn+2 − F2
Trying to prove: F1 + F2 + · · ·Fn = Fn+2 − 1
F1 = F3///− F2
F2 = F4///− F3///
F3 = F5///− F4///
F4 = F6///− F5///
......
Fn−1 = Fn+1////// − Fn///
+ Fn = Fn+2 − Fn+1//////
F1 + F2 + F3 + · · · + Fn = Fn+2 − 1
Trying to prove: F1 + F2 + · · · + Fn = Fn+2 − 1
We’ve just proved a theorem!!
Theorem
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F2 + F3 + · · · + Fn = Fn+2 − 1
An example
Recall the first several Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
What is this sum?
1+1+2+3+5+8+13+21+34+55+89+144 = 377−1 = 376!!
The “even” Fibonacci Numbers
What about the first few Fibonacci numbers with even index:
F2,F4,F6, . . . ,F2n, . . .
Let’s call them “even” Fibonaccis, since their index is even,although the numbers themselves aren’t always even!!
The “even” Fibonacci Numbers
Some notation: The first “even” Fibonacci number is F2 = 2.
The second “even” Fibonacci number is F4 = 3.
The third “even” Fibonacci number is F6 = 8.
The tenth “even” Fibonacci number is F20 =??.
The nth “even” Fibonacci number is F2n.
HOMEWORK
Tonight, try to come up with a formula for the sum of the first few“even” Fibonacci numbers.
