Where 1 radic52 is the golden ratio for n isin n define

Question

Math 104: Homework 1-

1. The Leonardo numbers are defined according to L₀ = 1, L₁ = 1, and L_n+1 = L_n + L_n-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 P_n to be the proposition that "both L_n = f(n) and L_n-1 = f(n - 1)". Apply mathematical induction to prove that P_n is true for all n ∈ N, and deduce that L_n = 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.