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Binet's Fibonacci Number Formula

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Binet's formula is a special case of the U_n Binet form with m=1, corresponding to the nth Fibonacci number,

F_n=(phi^n-(-phi)^(-n))/(sqrt(5))
(1)
=((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^nsqrt(5)),
(2)

where phi is the golden ratio. It was derived by Binet in 1843, although the result was known to Euler, Daniel Bernoulli, and de Moivre more than a century earlier.

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