Help proving this by induction?

Question

Consider a sport in which teams can score two types of goals, worth either 3 points or 7 points. For example, Team Miners might (theoretically speaking) score 32 points by accumulating, in succession, 3, 7, 3, 7, 3, 3, 3, and 3 points. Find the smallest possible n0 such that, for any n ≥ n0, a team can score exactly n points in a game. Prove your answer correct by strong induction.

Answer 1

First, find the number we think n0 should be, which it turns out is 12.

Next show three base cases so that we can always refer back three steps. That means n=12, 13, 14.

Next, using strong induction, let n >= 14 and assume everything from 12 to n satisfies the claim. Using this, show the claim is true for n+1 by splitting off a 3, and rearranging terms.