# How to find the closed form to the fibonacci numbers? [duplicate]

Possible Duplicate:
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.

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## marked as duplicate by J. M., Quixotic, robjohn♦, Martin Sleziak, t.b.Dec 12 '11 at 16:11

This question was marked as an exact duplicate of an existing question.

– deinst Dec 12 '11 at 15:46
It's on the Wikipedia page ^^. What search did you employ that failed to show you this? – The Chaz 2.0 Dec 12 '11 at 15:51

## 2 Answers

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

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

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But the OP asked how how to find the closed form. See J.M.'s dup link for some answers. – Bill Dubuque Dec 12 '11 at 16:03
Yes sorry but with my tablet is very difficult to post more involved answers. – 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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Perhaps you should clarify what you mean by an "expiated discussion". IMO the linked page leaves much to be desired. – Bill Dubuque Dec 12 '11 at 16:52