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 Fibonacci numbers! Let n any positive integer. If F n is what we use to describe the n th Fibonacci number, then F n = F n-1 + F n-2
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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!
Let n any positive integer. If Fn is what we use to describe the nth
Fibonacci number, then
Fn = Fn−1 + Fn−2
Trying to prove: F1 + F2 + · · · + Fn = Fn+2 − 1
We proved a theorem about the sum of the first few Fibonaccinumbers:
Theorem
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F2 + F3 + · · · + Fn = Fn+2 − 1
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 = 1.
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.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Let’s look at the first few:
F2 + F4 = 1 + 3 = 4
F2 + F4 + F6 = 1 + 3 + 8 = 12
F2 + F4 + F6 + F8 = 1 + 3 + 8 + 21 = 33
F2 + F4 + F6 + F8 + F10 = 1 + 3 + 8 + 21 + 55 = 88
See a pattern?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
It looks like the sum of the first few “even” Fibonacci numbers isone less than another Fibonacci number.
But which one?
F2 + F4 = 1 + 3 = 4 = F5 − 1
F2 + F4 + F6 = 1 + 3 + 8 = 12 = F7 − 1
F2 + F4 + F6 + F8 = 1 + 3 + 8 + 21 = 33 = F9 − 1
F2 + F4 + F6 + F8 + F10 = 1 + 3 + 8 + 21 + 55 = 88 = F11 − 1
Can we come up with a formula for the sum of the first few “even”Fibonacci numbers?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Let’s look at the last one we figured out:
F2 + F4 + F6 + F8 + F10 = 1 + 3 + 8 + 21 + 55 = 88 = F11 − 1
The sum of the first 5 even Fibonacci numbers (up to F10) is the11th Fibonacci number less one.
Maybe it’s true that the sum of the first n “even” Fibonacci’s isone less than the next Fibonacci number. That is,
Conjecture
For any positive integer n, the Fibonacci numbers satisfy:
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1
Trying to prove: F2 + F4 + · · ·F2n = F2n+1 − 1
Let’s try to prove this!!
We’re trying to find out what
F2 + F4 + · · ·F2n
is equal to.
Well, let’s proceed like the last theorem.
Trying to prove: F2 + F4 + · · ·F2n = F2n+1 − 1
We know F3 = F2 + F1.
So, rewriting this a little:
F2 = F3 − F1
Also: we know F5 = F4 + F3.
So:F4 = F5 − F3
In general:
Feven = Fnext odd − Fprevious odd
Let’s put these together:
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3 − F1
F4 = F5 − F3
F6 = F7 − F5
F8 = F9 − F7
......
F2n−2 = F2n−1 − F2n−3
F2n = F2n+1 − F2n−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: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5 − F3///
F6 = F7 − F5
F8 = F9 − F7
......
F2n−2 = F2n−1 − F2n−3
+ F2n = F2n+1 − F2n−1
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7 − F5///
F8 = F9 − F7
......
F2n−2 = F2n−1 − F2n−3
+ F2n = F2n+1 − F2n−1
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7///− F5///
F8 = F9 − F7///
......
F2n−2 = F2n−1 − F2n−3
+ F2n = F2n+1 − F2n−1
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7///− F5///
F8 = F9///− F7///
......
F2n−2 = F2n−1 − F2n−3////////
+ F2n = F2n+1 − F2n−1
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7///− F5///
F8 = F9///− F7///
......
F2n−2 = F2n−1////////− F2n−3////////
+ F2n = F2n+1 − F2n−1////////
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7///− F5///
F8 = F9///− F7///
......
F2n−2 = F2n−1////////− F2n−3////////
+ F2n = F2n+1 − F2n−1////////
F2 + F4 + F6 + · · · + F2n = F2n+1 − F1
Trying to prove: F2 + F4 + · · · + F2n = F2n+1 − 1
F2 = F3///− F1
F4 = F5///− F3///
F6 = F7///− F5///
F8 = F9///− F7///
......
F2n−2 = F2n−1////////− F2n−3////////
+ F2n = F2n+1 − F2n−1////////
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1
Trying to prove: F2 + F4 + · · ·F2n = F2n+1 − 1
This proves our second theorem!!
Theorem
For any positive integer n, the even Fibonacci numbers satisfy:
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1
The “odd” Fibonacci Numbers
What about the first few Fibonacci numbers with odd index:
F1,F3,F5, . . . ,F2n−1, . . .
Let’s call them “odd” Fibonaccis, since their index is odd,although the numbers themselves aren’t always odd!!
The “odd” Fibonacci Numbers
Some notation: The first “odd” Fibonacci number is F1 = 1.
The second “odd” Fibonacci number is F3 = 2.
The third “odd” Fibonacci number is F5 = 5.
The tenth “odd” Fibonacci number is F19 =??.
The nth “odd” Fibonacci number is F2n−1.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Let’s look at the first few:
F1 + F3 = 1 + 2 = 3
F1 + F3 + F5 = 1 + 2 + 5 = 8
F1 + F3 + F5 + F7 = 1 + 2 + 5 + 13 = 21
F1 + F3 + F5 + F7 + F9 = 1 + 2 + 5 + 13 + 34 = 55
See a pattern?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
It looks like the sum of the first few “odd” Fibonacci numbers isanother Fibonacci number.
But which one?
F1 + F3 = 1 + 2 = 3 = F4
F1 + F3 + F5 = 1 + 2 + 5 = 8 = F6
F1 + F3 + F5 + F7 = 1 + 2 + 5 + 13 = 21 = F8
F1 + F3 + F5 + F7 + F9 = 1 + 2 + 5 + 13 + 34 = 55 = F10
Can we come up with a formula for the sum of the first few “even”Fibonacci numbers?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Maybe it’s true that the sum of the first n “odd” Fibonacci’s is thenext Fibonacci number. That is,
Conjecture
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F3 + F5 + · · · + F2n−1 = F2n
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
Let’s try to prove this!!
We know F2 = F1.
So, rewriting this a little:
F1 = F2
Also: we know F4 = F3 + F2.
So:F3 = F4 − F2
In general:Fodd = Fnext even − Fprevious even
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
F1 = F2
F3 = F4 − F2
F5 = F6 − F4
F7 = F8 − F6
......
F2n−3 = F2n−2 − F2n−4
+ F2n−1 = F2n − F2n−2
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 + F3 + · · · + F2n−1 = F2n
F1 = F2///
F3 = F4 − F2///
F5 = F6 − F4
F7 = F8 − F6
......
F2n−3 = F2n−2 − F2n−4
+ F2n−1 = F2n − F2n−2
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
F1 = F2///
F3 = F4///− F2///
F5 = F6 − F4///
F7 = F8 − F6
......
F2n−3 = F2n−2 − F2n−4
+ F2n−1 = F2n − F2n−2
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
F1 = F2///
F3 = F4///− F2///
F5 = F6///− F4///
F7 = F8 − F6///
......
F2n−3 = F2n−2 − F2n−4
+ F2n−1 = F2n − F2n−2
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
F1 = F2///
F3 = F4///− F2///
F5 = F6///− F4///
F7 = F8///− F6///
......
F2n−3 = F2n−2 − F2n−4////////
+ F2n−1 = F2n − F2n−2
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
F1 = F2///
F3 = F4///− F2///
F5 = F6///− F4///
F7 = F8///− F6///
......
F2n−3 = F2n−2////////− F2n−4////////
+ F2n−1 = F2n − F2n−2////////
F1 + F3 + · · · + F2n−1 = F2n
Trying to prove: F1 + F3 + · · · + F2n−1 = F2n
This proves our third theorem!!
Theorem
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F3 + F5 + · · · + F2n−1 = F2n
So far:
Theorem (Sum of first few Fibonacci numbers.)
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F2 + F3 + · · · + Fn = Fn+2 − 1
Theorem (Sum of first few EVEN Fibonacci numbers.)
For any positive integer n, the Fibonacci numbers satisfy:
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1
Theorem (Sum of first few ODD Fibonacci numbers.)
For any positive integer n, the Fibonacci numbers satisfy:
F1 + F3 + F5 + · · · + F2n−1 = F2n
One more theorem about sums:
Theorem (Sum of first few SQUARES of Fibonacci numbers.)
For any positive integer n, the Fibonacci numbers satisfy:
F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
We’ll take advantage of the following fact:
For each Fibonacci number Fn
Fn+1 = Fn + Fn−1
Fn = Fn+1 − Fn−1
F 2n = Fn · Fn = Fn · (Fn+1 − Fn−1)
F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Let’s check this formula for a few different valus of n:
F 2n = Fn · Fn+1 − Fn · Fn−1
n = 4:F 2
4 = F4 · F4+1 − F4 · F4−1
F 24 = F4 · F5 − F4 · F3
32 = 3 · 5 − 3 · 2
9 = 15 − 6
It works for n = 4!
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Let’s check this formula for a few different valus of n:
F 2n = Fn · Fn+1 − Fn · Fn−1
n = 8:F 2
8 = F8 · F8+1 − F8 · F8−1
F 28 = F8 · F9 − F8 · F7
212 = 21 · 34 − 21 · 13
441 = 714 − 273 = 441
It works for n = 8!
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Let’s use this formula:
F 2n = Fn · Fn+1 − Fn · Fn−1
To find out what the sum of the first few SQUARES of Fibonaccinumbers is:
F 21 + F 2
2 + F 23 + · · · + F 2
n =????
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
We’re using:
F 2n = Fn · Fn+1 − Fn · Fn−1
F 21 = F1 · F2
F 22 = F2 · F3 − F2 · F1
F 23 = F3 · F4 − F3 · F2
F 24 = F4 · F5 − F4 · F3
......
F 2n−1 = Fn−1 · Fn − Fn−1 · Fn−2
+ F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
F 21 = F1 · F2/////////
F 22 = F2 · F3 − F2 · F1/////////
F 23 = F3 · F4 − F3 · F2
F 24 = F4 · F5 − F4 · F3
......
F 2n−1 = Fn−1 · Fn − Fn−1 · Fn−2
+ F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
F 21 = F1 · F2/////////
F 22 = F2 · F3/////////− F2 · F1/////////
F 23 = F3 · F4 − F3 · F2/////////
F 24 = F4 · F5 − F4 · F3
......
F 2n−1 = Fn−1 · Fn − Fn−1 · Fn−2
+ F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
F 21 = F1 · F2/////////
F 22 = F2 · F3/////////− F2 · F1/////////
F 23 = F3 · F4/////////− F3 · F2/////////
F 24 = F4 · F5 − F4 · F3/////////
......
F 2n−1 = Fn−1 · Fn − Fn−1 · Fn−2
+ F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
F 21 = F1 · F2/////////
F 22 = F2 · F3/////////− F2 · F1/////////
F 23 = F3 · F4/////////− F3 · F2/////////
F 24 = F4 · F5/////////− F4 · F3/////////
......
F 2n−1 = Fn−1 · Fn − Fn−1 · Fn−2///////////////
+ F 2n = Fn · Fn+1 − Fn · Fn−1
Trying to prove: F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
F 21 = F1 · F2/////////
F 22 = F2 · F3/////////− F2 · F1/////////
F 23 = F3 · F4/////////− F3 · F2/////////
F 24 = F4 · F5/////////− F4 · F3/////////
......
F 2n−1 = Fn−1 · Fn+1/////////////// − Fn−1 · Fn−2///////////////
+ F 2n = Fn · Fn+1 − Fn · Fn−1////////////
F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Our fourth theorem:
Theorem (Sum of first few SQUARES of Fibonacci numbers.)
For any positive integer n, the Fibonacci numbers satisfy:
F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
So far:
Theorem (Sum of first few Fibonacci numbers.)
F1 + F2 + F3 + · · · + Fn = Fn+2 − 1
Theorem (Sum of first few EVEN Fibonacci numbers.)
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1
Theorem (Sum of first few ODD Fibonacci numbers.)
F1 + F3 + F5 + · · · + F2n−1 = F2n
Theorem (Sum of first few SQUARES of Fibonacci numbers.)
F 21 + F 2
2 + F 23 + · · · + F 2
n = Fn · Fn+1
Examples
1 + 2 + 5 + 13 + 34 + 89 + 233 + 610 + 1597 + 4181 = 6765
1+1+4+9+25+64+169+441+1156+3025+7921 = 89·144 = 12816
Now, complete this worksheet .....
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