HW s 3-3 - MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1....

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Unformatted text preview: MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1. Define the Fibonacci sequence { f 1 , f 2 , f 3 , ... } by f 1 = f 2 = 1 and for n > 2, f n = f n- 1 + f n- 2 . Use induction to prove that for all natural numbers n , f n = n- n - , where = 1 + 5 2 and = 1- 5 2 . (Hint: and are the solutions to the equation x 2- x- 1 = 0.) Solution: The proof will proceed by induction. Assume that n = 1. Then f n = 1 and 1- 1 - = 1. Hence the result holds for n = 1. We also need to prove the result for n = 2 since the recursion formula for f n only holds for n > 2. So assume n = 2. Then f 2 = 1 and 2- 2 - = + = 1 + 5 2 + 1- 5 2 = 1 . Hence the result holds for n = 2. Let n N , n > 2 and assume that the result holds for all k { 1 , 2 , ..., n } . We must show that the result holds for n + 1. By the induction hypothesis and the definition of the Fibonacci numbers, f n +1 = f n + f n- 1 = n- n - + n- 1- n- 1 - = n + n- 1- ( n + n- 1 ) - ....
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HW s 3-3 - MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1....

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