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prove this fibonacci sequence w/o binet formula

skatenerd

Active member
Oct 3, 2012
114
I have a problem and honestly have no idea even where to start. I've been staring at it and thinking about it for over 24 hours...
Let \(u_n\) denote the \(n^{th}\) Fibonacci number. Without using the Binet formula for \(u_n\), prove the following for all natural numbers \(m\) and \(n\) with \(m\geq{2}\):
$$u_{m+n}=u_{m-1}u_n+u_mu_{n+1}$$

I have gone through a couple proofs regarding the Fibonacci numbers in this class before, but never one with two unknowns, and also never one where we weren't able to use the Binet formula. Without the Binet formula, what do I have to work off of? Do I need to write the derivation for the Binet formula within this whole proof?
 
Last edited:

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Try induction on $n$.