Fibonacci numbers and golden ratio

Prove that Fib(n) is the closest integer to φⁿ/√5 , where φ = (1 + √5)/2 . Hint: Let ψ = (1 - √5)/2 . Use induction and the definition of the Fibonacci numbers (see section 1.2.2) to prove that Fib(n) = (φⁿ - ψⁿ)/√5 .

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