Find a closed form for the recurrence an an-1 4an-2 -4an-3

Question

Question: 1. Find several perfect matchings of the Petersen graph.

2. Find a closed form for the recurrence a_n = a_n-1 +4a_n-2 -4a_n-3 and initial values a₀ = 4, a₁ = 2, a₂ = 10.

3. Consider a function on the real numbers defined by f(1) = q and f(a+b) = f(a)· f(b) for all real numbers a,b.

(a) Prove by induction that f(n) = q_n for all n ∈ N.

(b) Show that if q ≠ 0, then f(0) = 1.