Binet's Fibonacci Number Formula

Author

Eric W. Weisstein
January 21, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/BinetsFibonacciNumberFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Sums

Fibonacci[Range[10]]

(Fibonacci[n])==(((ϕ^n)-(((-ϕ))^(-n)))/(Sqrt[5]))

(1/(Sqrt[5]))Table[(GoldenRatio^n-(((-GoldenRatio))^(-n))),{n,10}]//FunctionExpand//FullSimplify

(Fibonacci[n])==((((((Sqrt[5])+1))^n)-(((1-(Sqrt[5])))^n))/((Sqrt[5]) (2^n)))

Table[(((((1+(Sqrt[5])))^n)-(((1-(Sqrt[5])))^n))/((2^n)(Sqrt[5]))),{n,10}]//Expand