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

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Finally We Will Investigate Some Properties Of The Fibonacci Numbers These Are Defined By F 1 F 1 And F Fn 1 Fn 1 (35.27 KiB) Viewed 38 times

Finally, we will investigate some properties of the Fibonacci numbers. These are defined by F₁=1, F₂= 1, and F₁ Fn-1+Fn-2 for n ≥ 3. So the first few are given by 1, 1, 2, 3, 5, 8,.... Let a = 18 and 3= 1; these are the two roots of the polynomial r²-x-1=0. Theorem 2. For all n E N we have F a"-3" 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 Fn-be? Example. Using the corollary, we find that Fio... use a computer for this Lemma 6. For all n EN we have 1 + F2 + F₁+ + F2n = F2n+1- Proof. ... 0 O