Problem 5: Define the sequence of rational numbers (an)>1 by a1 = 0, 02 = 1, and On := On-1 Van-2 for n > 3. (i) Show that an anti for all n 2 1. (ii) Compute a few terms of the sequence. Show that for n 2 1, we have On = Fn-1/Fn with Fn-1 and F, relatively prime, where F, is the n-th Fibonacci number. (iii) For n 2 1, show that FntiFn-1 - F2 = +1. Hint: use the previous parts] (iv) You may assume that the sequence (On) converges to a real number o (this is not too hard to show). Prove that a = (-1 + v5)/2. Deduce that lim Fn+1 1 + v5 n-+00 Fn 2 [Hint: first show that On + 1 = 1/on+1 and take limits of both sides.]