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ASSIGNMENT 2.1 There are two parts to this assignment. Part A is to be completed online before noon on Friday, January 14. Part B, which should be typeset (preferably with LaTeX), is to be submitted to the assignment box by the same time. Include your full name and student ID, along with the subject and assignment number, at the top of the front page. Part A [15 marks] This part of the assignment can be found online, labelled Assignment 2.1, at www.mathxl.com. Questions about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed to him. To ask me questions, email me directly or come to my office. Part B [15 marks] The Fibonacci sequence {an } is defined by an = 1 an−1 + an−2 for n = 1, 2 . for n ≥ 3 Let {rn } be the sequence of ratios defined by rn = an+1 /an . In this question, you will prove that {rn } √ converges to ϕ, the golden ratio (1 + 5)/2. (i) Verify the identities rn = 1 + 1 for n ≥ 2 rn−1 and rn = 2 − 1 for n ≥ 3. rn−2 + 1 (ii) Prove that if {rn } converges, it converges to ϕ. (iii) Show that if rn−2 ≥ ϕ, then rn ≥ ϕ; and if rn−2 ≤ ϕ, then rn ≤ ϕ. (iv) Prove, by induction or otherwise, that the sequences {r2n−1 } and {r2n } are increasing and decreasing, respectively. (v) Prove that the assumption in part (b) is justified; that is, prove that {rn } converges.