Please answer questions

1. (a) Prove that the Fibonacci sequence is also the solution to the recurrence relation an = 5an-4 + 3an-5, with initial conditions ao = 0, a1, = 1, a2 = 1, a3 = 2, a4 = 3. (b) Show that a term Fn of the Fibonacci sequence is divisible by 5 if and only if n is divisible by 5. 2. Let Fr denote the kth term of the Fibonacci sequence. Give a combi- natorial proof that, for all m _ 1 and n 2 0, one has Fmon = Fm-1Fn + FmFn+1. Hint: Recall from the second video this week that these numbers also count certain tilings of a 1 x n grid