Prove that any postage greater than 17 can be made using 4 and 7 cent stamps

Question

Please use strong induction for the problem. I know that regular induction doesn't work. I assume there is a proof by logic by simply saying that 18, 19 and 20 cents can be made using these stamps and that you just need to subtract a 7 cent stamp and then add two four cent stamps to get the next higher cent value. However, that is not formaly strong induction.

Answer 1

Hint 1 $$18=7+7+4 \\ 19=7+4+4+4\\ 20=4+4+4+4+4 \\ 21=7+7+7$$

Hint 2: If $n \geq 21$ and you know that $P(1), P(2),.., P(n)$ is true, then $P(n-4)$ is true.

Answer 2

Hint: If you can make any four consecutive denominations you can make any afterward by adding four-cent stamps. So all you have to do is show combinations adding up to 18 to 21 cents.