The Fibonacci Numbers

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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 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

Trying to prove: F 1 + F 2 + + F n = F n+2 1 We proved a theorem about the sum of the first few Fibonacci numbers: Theorem For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 2 + F 3 + + F n = F n+2 1

The even Fibonacci Numbers What about the first few Fibonacci numbers with even index: F 2, F 4, F 6,, F 2n, 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 F 2 = 1 The second even Fibonacci number is F 4 = 3 The third even Fibonacci number is F 6 = 8 The tenth even Fibonacci number is F 20 =?? The n th even Fibonacci number is F 2n

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: F 2 + F 4 = 1 + 3 = 4 F 2 + F 4 + F 6 = 1 + 3 + 8 = 12 F 2 + F 4 + F 6 + F 8 = 1 + 3 + 8 + 21 = 33 F 2 + F 4 + F 6 + F 8 + F 10 = 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 is one less than another Fibonacci number But which one? F 2 + F 4 = 1 + 3 = 4 = F 5 1 F 2 + F 4 + F 6 = 1 + 3 + 8 = 12 = F 7 1 F 2 + F 4 + F 6 + F 8 = 1 + 3 + 8 + 21 = 33 = F 9 1 F 2 + F 4 + F 6 + F 8 + F 10 = 1 + 3 + 8 + 21 + 55 = 88 = F 11 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: F 2 + F 4 + F 6 + F 8 + F 10 = 1 + 3 + 8 + 21 + 55 = 88 = F 11 1 The sum of the first 5 even Fibonacci numbers (up to F 10 ) is the 11 th Fibonacci number less one Maybe it s true that the sum of the first n even Fibonacci s is one less than the next Fibonacci number That is, Conjecture For any positive integer n, the Fibonacci numbers satisfy: F 2 + F 4 + F 6 + + F 2n = F 2n+1 1

Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 Let s try to prove this!! We re trying to find out what is equal to F 2 + F 4 + F 2n Well, let s proceed like the last theorem

Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 We know F 3 = F 2 + F 1 So, rewriting this a little: F 2 = F 3 F 1 Also: we know F 5 = F 4 + F 3 So: F 4 = F 5 F 3 In general: Let s put these together: F even = F next odd F previous odd

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = F 3 F 1 F 4 = F 5 F 3 F 6 = F 7 F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 F 2n = F 2n+1 F 2n 1 Adding up all the terms on the left sides will give us something equal to the sum of the terms on the right sides

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = F 5 /// F 3 F 6 = F 7 F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = F 7 /// F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = F 9 /// F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 F 2n 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1 F 2 + F 4 + F 6 + + F 2n = F 2n+1 F 1

Trying to prove: F 2 + F 4 + + F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1 F 2 + F 4 + F 6 + + F 2n = F 2n+1 1

Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 This proves our second theorem!! Theorem For any positive integer n, the even Fibonacci numbers satisfy: F 2 + F 4 + F 6 + + F 2n = F 2n+1 1

The odd Fibonacci Numbers What about the first few Fibonacci numbers with odd index: F 1, F 3, F 5,, F 2n 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 F 1 = 1 The second odd Fibonacci number is F 3 = 2 The third odd Fibonacci number is F 5 = 5 The tenth odd Fibonacci number is F 19 =?? The n th odd Fibonacci number is F 2n 1

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Let s look at the first few: F 1 + F 3 = 1 + 2 = 3 F 1 + F 3 + F 5 = 1 + 2 + 5 = 8 F 1 + F 3 + F 5 + F 7 = 1 + 2 + 5 + 13 = 21 F 1 + F 3 + F 5 + F 7 + F 9 = 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 is another Fibonacci number But which one? F 1 + F 3 = 1 + 2 = 3 = F 4 F 1 + F 3 + F 5 = 1 + 2 + 5 = 8 = F 6 F 1 + F 3 + F 5 + F 7 = 1 + 2 + 5 + 13 = 21 = F 8 F 1 + F 3 + F 5 + F 7 + F 9 = 1 + 2 + 5 + 13 + 34 = 55 = F 10 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 the next Fibonacci number That is, Conjecture For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F 5 + + F 2n 1 = F 2n

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n Let s try to prove this!! We know F 2 = F 1 So, rewriting this a little: F 1 = F 2 Also: we know F 4 = F 3 + F 2 So: F 3 = F 4 F 2 In general: F odd = F next even F previous even

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = F 2 F 3 = F 4 F 2 F 5 = F 6 F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2 Adding up all the terms on the left sides will give us something equal to the sum of the terms on the right sides

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = /// F 2 F 3 = F 4 /// F 2 F 5 = F 6 F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = F 6 /// F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = F 8 /// F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = /// F 8 /// F 6 F 2n 3 = F 2n 2 //////// F 2n 4 + F 2n 1 = F 2n F 2n 2

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = /// F 8 /// F 6 F 2n 3 = //////// F 2n 2 //////// F 2n 4 + F 2n 1 = F 2n //////// F 2n 2 F 1 + F 3 + + F 2n 1 = F 2n

Trying to prove: F 1 + F 3 + + F 2n 1 = F 2n This proves our third theorem!! Theorem For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F 5 + + F 2n 1 = F 2n

So far: Theorem (Sum of first few Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 2 + F 3 + + F n = F n+2 1 Theorem (Sum of first few EVEN Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 2 + F 4 + F 6 + + F 2n = F 2n+1 1 Theorem (Sum of first few ODD Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F 5 + + F 2n 1 = F 2n

One more theorem about sums: Theorem (Sum of first few SQUARES of Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 We ll take advantage of the following fact: For each Fibonacci number F n F n+1 = F n + F n 1 F n = F n+1 F n 1 F 2 n = F n F n = F n (F n+1 F n 1 ) F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 Let s check this formula for a few different valus of n: F 2 n = F n F n+1 F n F n 1 n = 4: F 2 4 = F 4 F 4+1 F 4 F 4 1 F 2 4 = F 4 F 5 F 4 F 3 3 2 = 3 5 3 2 9 = 15 6 It works for n = 4!

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 Let s check this formula for a few different valus of n: F 2 n = F n F n+1 F n F n 1 n = 8: F 2 8 = F 8 F 8+1 F 8 F 8 1 F 2 8 = F 8 F 9 F 8 F 7 21 2 = 21 34 21 13 441 = 714 273 = 441 It works for n = 8!

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 Let s use this formula: F 2 n = F n F n+1 F n F n 1 To find out what the sum of the first few SQUARES of Fibonacci numbers is: F 2 1 + F 2 2 + F 2 3 + + F 2 n =????

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 We re using: F 2 n = F n F n+1 F n F n 1 F1 2 = F 1 F 2 F2 2 = F 2 F 3 F 2 F 1 F3 2 = F 3 F 4 F 3 F 2 F4 2 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = F 2 F 3 ///////// F 2 F 1 F 2 3 = F 3 F 4 F 3 F 2 F 2 4 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = F 3 F 4 ///////// F 3 F 2 F 2 4 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = F 4 F 5 ///////// F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = ///////// F 4 F 5 ///////// F 4 F 3 Fn 1 2 = F n 1 F n F/////////////// n 1 F n 2 + F 2 n = F n F n+1 F n F n 1

Trying to prove: F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = ///////// F 4 F 5 ///////// F 4 F 3 Fn 1 2 = F/////////////// n 1 F n+1 F/////////////// n 1 F n 2 + Fn 2 = F n F n+1 //////////// F n F n 1 F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+1

Our fourth theorem: Theorem (Sum of first few SQUARES of Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F1 2 + F2 2 + F3 2 + + Fn 2 = F n F n+1

So far: Theorem (Sum of first few Fibonacci numbers) F 1 + F 2 + F 3 + + F n = F n+2 1 Theorem (Sum of first few EVEN Fibonacci numbers) F 2 + F 4 + F 6 + + F 2n = F 2n+1 1 Theorem (Sum of first few ODD Fibonacci numbers) F 1 + F 3 + F 5 + + F 2n 1 = F 2n Theorem (Sum of first few SQUARES of Fibonacci numbers) F 2 1 + F 2 2 + F 2 3 + + F 2 n = F n F n+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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