Finally, we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and FnF-1+ F

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Finally We Will Investigate Some Properties Of The Fibonacci Numbers These Are Defined By F 1 F 1 And Fnf 1 F 1 (35.6 KiB) Viewed 36 times

Finally, we will investigate some properties of the Fibonacci numbers. These are defined by F₁ = 1, F₂ = 1, and FnF-1+ Fn-2 for n ≥ 3. So the first few are given by 1,1,2,3,5,8,.... Let a = ¹+5 and 8 = 1; these are the two roots of the polynomial x²-x-1=0. Theorem 2. For all ne N we have F₁ = 7 Proof. Hint: this should involve no unpleasant algebra. Use the polynomial. Corollary 1. For all n EN, F, is the closest integer to Proof. Hint: How big can F₁-be? Example. Using the corollary, we find that Fio ........ use a computer for this Lemma 6. For all neN we have 1+F2+ F₁+ + F2n = F2n+1. Proof. =