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Problem 5. Consider the Fibonacci numbers, define recursively by Fo=0, F₁=1, and Fn = Fn-1 + Fn-2 for all n ≥ 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, . For all n ≥ 2, define the rational number rn by the fraction 1 2 3 5 8 1' 2' 3'5' Fn -; so the first few terms are Fn-1 (a) Prove that for all n ≥4, we have rn = Tn-1 V rn-2. (b) Prove that the sequence rn converges (to a real number). (c) Prove that rn converges to the golden ratio: 1+√5 2 For this problem, you can use any result that you may have seen in your Calculus classes.

Problem 5. Consider the Fibonacci numbers, define recursively by Fo=0, F₁=1, and Fn = Fn-1 + Fn-2 for all n ≥ 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, . For all n ≥ 2, define the rational number rn by the fraction 1 2 3 5 8 1' 2' 3'5' Fn -; so the first few terms are Fn-1 (a) Prove that for all n ≥4, we have rn = Tn-1 V rn-2. (b) Prove that the sequence rn converges (to a real number). (c) Prove that rn converges to the golden ratio: 1+√5 2 For this problem, you can use any result that you may have seen in your Calculus classes.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 5. Consider the Fibonacci numbers, define recursively by Fo= 0, F₁ = 1, and Fn = Fn-1 + Fn-2 for all n> 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13,.. For all n 2, define the rational number rn by the fraction 12358 1'1'2' 3'5 Fn F 6 = (a) Prove that for all n ≥ 4, we have rn = Tn-1 V rn-2. (b) Prove that the sequence în converges (to a real number). (c) Prove that rn converges to the golden ratio: -; so the first few terms are 'n-1 1+√5 2 For this problem, you can use any result that you may have seen in your Calculus classes.
Transcribed Image Text:Problem 5. Consider the Fibonacci numbers, define recursively by Fo= 0, F₁ = 1, and Fn = Fn-1 + Fn-2 for all n> 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13,.. For all n 2, define the rational number rn by the fraction 12358 1'1'2' 3'5 Fn F 6 = (a) Prove that for all n ≥ 4, we have rn = Tn-1 V rn-2. (b) Prove that the sequence în converges (to a real number). (c) Prove that rn converges to the golden ratio: -; so the first few terms are 'n-1 1+√5 2 For this problem, you can use any result that you may have seen in your Calculus classes.
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