For this problem I have to prove using two different types of induction to show that using only 3 cent stamps and 5 cent stamps, any postage amount 8 cents or greater can be formed. Using the two styles of induction. I'm not sure if what I have is correct could someone please help me out?
I think I've done the one with the strong induction, which has multiple base cases before the induction step. That is if it's correct.
Ex: Base cases: $$P(8) = 8 = 3 +5,\\ P(9) = 9 = 3*3 + 5(0),\\ P(10) = 10 = 3(0) + 5(2)$$
Induction step: assume $n$ is greater than $10$, then $n-3$ greater than or equal to $8$ is true. Let $a'$ and $b'$ such that $n-3 = 3a' + 5b'$
$$n = 3+ (n-3)\\ = 3 + 3a' + 5b\\ = 3(a' + 1) +5b'$$
I'm not sure if this is correct or not.
Since there isn't a LHS or RHS how would I prove this using another form of induction?