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Unformatted text preview: Fall 2009 CS131 – Combinatorial Structures Homework 9 Homework 9, due Nov 24 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1. (10pts) Recall the equation ϕ = 1 + 1/ ϕ for the solution of the equation for the golden section. Suppose that instead of using the formula for solving a second-order equation, we will try to approximate a solution, using the following recursive sequence: x 1 = 1, x n + 1 = 1 + 1 x n . We get the sequence 1,2,3/2,5/3,.... Show x n = F n + 1 F n . Solution. We will use induction. The relation holds for n = 1. Assume it holds for n , we show it for n + 1: x n + 1 = 1 + 1/ x n = 1 + F n F n + 1 = F n + 1 + F n F n + 1 = F n + 2 F n + 1 , which is the statement for n + 1. Problem 2. (10pts) Using the notation of the previous problem, prove the following: x n + 1- x n = 1 F n F n + 1 if n is odd, and- 1 F n F n + 1 if it is even....
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- Fall '11