Question: Problem 4. The nth Fibonacci number F is defined by Fo= 0, F = 1, and Fn+2 = Fn+1+F, 80 F = 1 F3

Problem 4. The nth Fibonacci number F is defined by Fo= 0,

Problem 4. The nth Fibonacci number F is defined by Fo= 0, F = 1, and Fn+2 = Fn+1+F, 80 F = 1 F3 = 2, F3, F5 = 5, etc. i. Prove that ged(Fn+1, Fn) = 1 for all n. ii. Solve Fn+12+ Fay =1 with r, y Z (for all n). iii. Prove that ged(Fn+2, Fn) = 1 for all n, and gcd(Fn+3+ Fn) = 1 or 2. iv. Prove that ged(Fn+k+ Fn) divides F for all n.

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solution iWe can prove this statement by induction on n Base Case n 0 F1 1 and F0 0 so gcdF1 F0 1 Inductive Step assume gcdFn1 Fn 1 for some n We must ... View full answer

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