Problem 1. Consider the Fibonacci numbers, define recursively by F = 0, F = 1, and...

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Problem 1. Consider the Fibonacci numbers, define recursively by F = 0, F = 1, and Fn = Fn1 + Fn-2 for all n 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, .... Fn For all n >2, define the rational number rn by the fraction -; so the first few terms are Fn-1 1 2 3 5 8 1'1'2'3'5' (a) (5 pts) Prove that for all n 4, we have n = rn-1 V rn-2. (b) (5pts) Prove that the sequence n converges (to a real number). (c) (5 pts) Prove that converges to the golden ratio: 1+5 2 Problem 1. Consider the Fibonacci numbers, define recursively by F = 0, F = 1, and Fn = Fn1 + Fn-2 for all n 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, .... Fn For all n >2, define the rational number rn by the fraction -; so the first few terms are Fn-1 1 2 3 5 8 1'1'2'3'5' (a) (5 pts) Prove that for all n 4, we have n = rn-1 V rn-2. (b) (5pts) Prove that the sequence n converges (to a real number). (c) (5 pts) Prove that converges to the golden ratio: 1+5 2