The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci...
Transcript of The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci...
![Page 1: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/1.jpg)
The Fibonacci Numbers
A sequence of numbers (actually, positive integers) that occurmany times in
nature,
art,
music,
geometry,
mathematics,
economics,
and more....
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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
![Page 3: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/3.jpg)
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,
![Page 4: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/4.jpg)
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,
![Page 5: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/5.jpg)
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,
![Page 6: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/6.jpg)
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, . . .
![Page 7: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/7.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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:
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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?
![Page 21: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/21.jpg)
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?
![Page 22: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/22.jpg)
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:
![Page 23: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/23.jpg)
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:
![Page 24: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/24.jpg)
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:
![Page 25: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/25.jpg)
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:
![Page 26: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/26.jpg)
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:
![Page 27: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/27.jpg)
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:
![Page 28: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/28.jpg)
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:
![Page 29: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/29.jpg)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
![Page 30: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/30.jpg)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
![Page 31: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/31.jpg)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Another pinecone:
Find and count the number of spirals:
![Page 32: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/32.jpg)
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:
![Page 33: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/33.jpg)
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:
![Page 34: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/34.jpg)
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:
![Page 35: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/35.jpg)
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:
![Page 36: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/36.jpg)
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:
![Page 37: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/37.jpg)
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:
![Page 38: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/38.jpg)
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:
![Page 39: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/39.jpg)
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:
![Page 40: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/40.jpg)
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:
![Page 41: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/41.jpg)
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:
![Page 42: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/42.jpg)
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:
![Page 43: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/43.jpg)
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
![Page 44: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/44.jpg)
The Fibonacci Numbers
For example, let’s look at the sum of the first several of thesenumbers.
What is F1 + F2 + F3 + F4 · · · + Fn = ???
![Page 45: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/45.jpg)
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?
![Page 46: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/46.jpg)
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!
![Page 47: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/47.jpg)
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!
![Page 48: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/48.jpg)
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.)
![Page 49: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/49.jpg)
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
![Page 50: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/50.jpg)
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.
![Page 51: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/51.jpg)
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
![Page 52: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/52.jpg)
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
![Page 53: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/53.jpg)
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
![Page 54: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/54.jpg)
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
![Page 55: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/55.jpg)
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
![Page 56: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/56.jpg)
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//////
![Page 57: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/57.jpg)
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
![Page 58: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/58.jpg)
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
![Page 59: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/59.jpg)
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
![Page 60: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/60.jpg)
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!!
![Page 61: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/61.jpg)
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!!
![Page 62: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/62.jpg)
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.
![Page 63: The Fibonacci Numbers - Sam Houston State Universityldg005/data/mth164/F1.pdf · The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7094267f8b9aa2538c3722/html5/thumbnails/63.jpg)
HOMEWORK
Tonight, try to come up with a formula for the sum of the first few“even” Fibonacci numbers.