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how to derive binets formula. F n 1 5 ϕ n ϕ n where ϕ 1 2 1 5 is the golden ratio. Typically the formula is proven as a special case of a more general study of sequences in number theory.
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1 Verifying the Binet formula satisfies the recursion relation. The explicit formula for the terms of the Fibonacci sequence F n 1 5 2 n 1 5 2 n 5. The solutions of the characteristic equationx2-x-10are.
Binets formula expresses the n th Fibonacci number in terms of n and the golden ratio and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
Binets Formula for the nth Fibonacci number. A simple proof that Fib n Phi n Phi n5. This page contains two proofs of the formula for the Fibonacci numbers. Fibonacci numbers are strongly related to the golden ratio.
