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Prove this formula for the Fibonacci Sequence

How to find the closed form to the fibonacci numbers?

I have seen is possible calculate the fibonacci numbers without recursion, but, how can I find this formula? Where it come from?

Appreciate helps, thx.


marked as duplicate by J. M. is a poor mathematician, Quixotic, robjohn, Martin Sleziak, t.b. Dec 12 '11 at 16:11

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ See en.wikipedia.org/wiki/… $\endgroup$ – deinst Dec 12 '11 at 15:46
  • $\begingroup$ It's on the Wikipedia page ^^. What search did you employ that failed to show you this? $\endgroup$ – The Chaz 2.0 Dec 12 '11 at 15:51

The n-th Fibonacci number is given in closed form by

$$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n- \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n $$

  • 5
    $\begingroup$ But the OP asked how how to find the closed form. See J.M.'s dup link for some answers. $\endgroup$ – Bill Dubuque Dec 12 '11 at 16:03
  • $\begingroup$ Yes sorry but with my tablet is very difficult to post more involved answers. $\endgroup$ – Jon Dec 12 '11 at 20:01

This is probably the most expiated discussion of $n$-th term of Fibonacci series in world wide web.

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    $\begingroup$ Perhaps you should clarify what you mean by an "expiated discussion". IMO the linked page leaves much to be desired. $\endgroup$ – Bill Dubuque Dec 12 '11 at 16:52

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