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

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Fill in the P (X=x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -5, 1, 4, 5, and 6. Value x of X P ( X = x) -5 0.12 1 E 4 5 6 X e 0.22 0 0.25 ?