Binet's Formula

A special case of the $U_n$ Binet Form with $m=1$, corresponding to the $n$th Fibonacci Number,

$$F_n = {(1+\sqrt{5}\,)^n-(1-\sqrt{5}\,)^n\over 2^n\sqrt{5}}.$$

It was derived by Binet in 1843, although the result was known to Euler and to Daniel Bernoulli more than a century earlier.

See also Binet Forms, Fibonacci Number