Fibonacci numbers were defined as follows: f0 = 0, f1 = 1 andfor all n ≥ 1, fn+1 = fn + fn−1. i. For any natural number n ≥ 3,show that the quotient of fn+1 fn is 1 and the remainder of fn+1 fnis fn−1. ii. Prove (by induction/recursion, if you like) that theEuclidean Algorithm takes n − 2 loops to calculate gcd(fn, fn−1),for any integer n ≥ 3.
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