Lucas/Fibonacci Proof By Induction
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Lucas/Fibonacci Proof By Induction

[From: ] [author: ] [Date: 12-10-07] [Hit: ]
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Hey guys,

I've been having so many troubles with this problem and I'm hoping that you could give me some guidance. The problem is to prove the following by induction:

F(2n) = F(n)*L(n)

Here, F(n) is the Fibonacci Sequence and L(n) is the Lucas Sequence. I have about 3 paths I've gone down and none of them end well. If it helps you, I have a side proof to show that L(n) = F(n-1) + F(n+1), so that can be used if needed.

I appreciate any help you can give with this problem.

Thanks!

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'll assume that F(2n) = F^2(n+1) - F^2(n-1).

Then, this is easy:
F(2n) = F^2(n+1) - F^2(n-1)
.........= (F(n+1) - F(n-1)) * (F(n+1) + F(n-1))
.........= F(n) * (F(n+1) + F(n-1)), via Fibonacci recurrence
.........= F(n) * L(n), using your 'side relation'.
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Link to a proof of F(2n) = F^2(n+1) - F^2(n-1):
http://ph.answers.yahoo.com/question/ind…

I hope this helps!
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keywords: Fibonacci,Induction,Lucas,By,Proof,Lucas/Fibonacci Proof By Induction
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