Define a sequence a1, a2, a3, ... as follows: a1 = 1, a2 = 3, and ak = ak-1 + ak-2 for all...

Define a sequence a1, a2, a3, ... as follows: a1 = 1, a2 = 3, and ak = ak-1 + ak-2 for all integers k ≥ 3. (This sequence is known as the Lucas sequence.) Use strong mathematical induction to prove that  for all integers n ≥ 1.