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March 31st, 2014, 04:44 AM   #1
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Question Mathematical Induction, Fibonacci Sequence and Pascal's triangle

Hello guys,

During the creation of my math work on the topic of Pascal's Triangle I came up with a formula for counting elements of Fibonacci Sequence using the entries from Pascal's Triangle (binomial coefficients). I know that there is a general formula for that, which I have, explained it and proven by induction, but what I also wanted to do in my work, was to create two formulas for counting even entries of Fibonacci Sequence and the odd ones. What I am struggling with however, is how to prove it using the induction. I attach the screenshot of the page that deals directly with ODD numbers. If you guys could help me with the induction, I would greatly appreciate it.

I am looking forward to hearing from you!

Kind regards,

kbeski
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File Type: jpg Screen Shot 2014-03-31 at 2.43.38 PM.jpg (13.2 KB, 8 views)
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March 31st, 2014, 06:07 AM   #2
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I see that the picture I have uploaded is in a very small resolution, so here is the full size:

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March 31st, 2014, 09:31 AM   #3
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Proving (effectively by induction) that the right-hand sides are Fibonacci numbers is done in many websites.

The left-hand sides are sums of entries along certain diagonals of Pascal's triangle. Their additive property (as required for Fibonacci numbers) follows easily from the additive property used in the construction of Pascal's triangle.

In both cases, it is easier to deal with all the Fibonacci numbers instead of the even position and odd position ones separately.
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March 31st, 2014, 09:48 AM   #4
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Quote:
Originally Posted by skipjack View Post
Proving (effectively by induction) that the right-hand sides are Fibonacci numbers is done in many websites.

The left-hand sides are sums of entries along certain diagonals of Pascal's triangle. Their additive property (as required for Fibonacci numbers) follows easily from the additive property used in the construction of Pascal's triangle.

In both cases, it is easier to deal with all the Fibonacci numbers instead of the even position and odd position ones separately.
Hello skipjack,

I would like to thank you for your answer, but also tell you that I still have troubles with rearranging the formula in order to prove it.

The below is what I have already written as it comes to a general formula for obtaining any Fibonacci Number using Pascal's Triangle.



I honestly and sincerely have no idea how to continue it and I think I have already searched most of the internet to find a solution to my problem.

I will greatly appreciate your help
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March 31st, 2014, 10:33 AM   #5
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This article explains visually why the appropriate diagonals of Pascal's triangle generate the Fibonacci numbers. Do you understand it?
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March 31st, 2014, 10:40 AM   #6
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Yes I indeed understand it, but I am afraid it does not help with my question at all.
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March 31st, 2014, 11:15 AM   #7
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Binet's formula for the Fibonacci numbers is explained and proved here. Do you also understand that explanation?

For each article, the proof given can readily be converted to a formal proof by induction.

You also need an inductive proof that Pascal's triangle contains all the binomial coefficients, which is easily accomplished.

Last edited by skipjack; March 31st, 2014 at 11:21 AM.
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March 31st, 2014, 12:52 PM   #8
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skipjack, I do understand it, but I think that you don't get what my problem is. I am struggling with expanding the RHS from the last equation of the picture I have uploaded. I just don't know how to rearrange it in such manner, that I could be able to continue proving the validity of the formula.
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