Use mathematical induction (and proof by division into cases) to show that any postage of at least 12 cents can be obtained using 3 cent and 7 cent stamps.
So for this I understand that it can be solved using
induction without a strong hypothesis.
$$ n = 12 : 3*4 + 7*0$$ $$ n = 13 : 3*2 + 7*1$$ $$ n = 14 : 3*0 + 7*2$$
Induction hypothesis: Assume for some $k>14$ that $k = 3a + 7b$ for $a, b \in Z$.
Induction step: We will show that $k + 1$ can be made up of 3 and 7 cent stamps.
$$k+1 = 3a + 7b + 1$$ $$k+1 = 3a + 7b + 7 -6$$ $$k+1 = 3a - 6 + 7b + 7$$ $$k+1 = 3(a-2) + 7(b+1)$$
So when $a > 2$ we can make $k + 1$ stamps.
Is it at this point that I need to also clarify when $a = 1$ and $a = 0$ that other cases of $k$ will hold as well? Have I already proven my original statement? Or is this covered because I have shown the pattern in the base cases?