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we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. So the first few are given by 1, 1, 2, 3, 5, 8,… Let a = polynomial x²-x-1=0. Theorem 1 For all n N we have Fn = "-". Hint: this should involve no unpleasant algebra. Use the polynomial. Corollary 1 For all n EN, F₁ is the closest integer to 1+√5 and 3 = 1-5; these are the two roots of the Hint: How big can F₁-be? Using the corollary, we find that Fio Lemma 1 For all n EN we have 1+F₂ +F4 + + F2n = F2n+1. ******** use a computer for this

we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. So the first few are given by 1, 1, 2, 3, 5, 8,… Let a = polynomial x²-x-1=0. Theorem 1 For all n N we have Fn = "-". Hint: this should involve no unpleasant algebra. Use the polynomial. Corollary 1 For all n EN, F₁ is the closest integer to 1+√5 and 3 = 1-5; these are the two roots of the Hint: How big can F₁-be? Using the corollary, we find that Fio Lemma 1 For all n EN we have 1+F₂ +F4 + + F2n = F2n+1. ******** use a computer for this

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
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we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. So the first few are given by 1, 1, 2, 3, 5, 8,.... Let a = 1+√5 and 3 polynomial x²-x-1=0. Theorem 1 For all n N we have Fn = "-". Hint: this should involve no unpleasant algebra. Use the polynomial. Corollary 1 For all n EN, F, is the closest integer to 1-5; these are the two roots of the Hint: How big can F₁-be? Using the corollary, we find that F10 = Lemma 1 For all n EN we have 1+F2 +F4 + + F2n = F2n+1. ******** use a computer for this
Transcribed Image Text:we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. So the first few are given by 1, 1, 2, 3, 5, 8,.... Let a = 1+√5 and 3 polynomial x²-x-1=0. Theorem 1 For all n N we have Fn = "-". Hint: this should involve no unpleasant algebra. Use the polynomial. Corollary 1 For all n EN, F, is the closest integer to 1-5; these are the two roots of the Hint: How big can F₁-be? Using the corollary, we find that F10 = Lemma 1 For all n EN we have 1+F2 +F4 + + F2n = F2n+1. ******** use a computer for this
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