Difference between revisions of "Binet's formula"

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(Created page with "==Theorem== The following formula holds: $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ where $F_n$ denotes a Fibonacci number and $\phi$ denotes th...")
 
 
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The following formula holds:
 
The following formula holds:
 
$$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$
 
$$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$
where $F_n$ denotes a [[Fibonacci sequence|Fibonacci number]] and $\phi$ denotes the [[golden ratio]].  
+
where $F_n$ denotes a [[Fibonacci numbers|Fibonacci number]] and $\phi$ denotes the [[golden ratio]].  
  
 
==Proof==
 
==Proof==
  
 
==References==
 
==References==
 +
* {{PaperReference|On a General Fibonacci Identity|1965|John H. Halton|prev=Fibonacci numbers|next=F(-n)=(-1)^(n+1)F(n)}}
 +
* {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci zeta function|next=Fibonacci zeta in terms of a sum of binomial coefficients}}
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 22:32, 25 May 2017

Theorem

The following formula holds: $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ where $F_n$ denotes a Fibonacci number and $\phi$ denotes the golden ratio.

Proof

References