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Math 104: Homework 1-
1. The Leonardo numbers are defined according to L0 = 1, L1 = 1, and Ln+1 = Ln + Ln-1 + 1 for all n ∈ N. Consider the function
f(n) = 2/√5(?n+1 - (1 - ?)n+1) - 1
where ? = (1 +√5)/2 is the Golden Ratio. For n ∈ N, define Pn to be the proposition that "both Ln = f(n) and Ln-1 = f(n - 1)". Apply mathematical induction to prove that Pn is true for all n ∈ N, and deduce that Ln = f(n) for all n ∈ N ∪ {0}.
2. Show that √2 + √3 is irrational.
3. Show that ||a| - |b|| ≤ |a - b| for all a, b ∈ R.
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