HW s 3-3

# HW s 3-3 - MATH 290 – WRITING ASSIGNMENT 3 – SOLUTIONS...

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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 = α n- β n α- β , where α = 1 + √ 5 2 and β = 1- √ 5 2 . (Hint: α and β 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 α 1- β 1 α- β = 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 α 2- β 2 α- β = α + β = 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 = α n- β n α- β + α n- 1- β n- 1 α- β = α n + α n- 1- ( β n + β n- 1 ) α- β ....
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## This note was uploaded on 03/05/2011 for the course MATH 290 taught by Professor Alligood,k during the Fall '08 term at George Mason.

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

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