I proved that any amount greater than 24 cents can be obtained by a combination of 5 cents and 7 cents using strong induction. First I proved the following base cases:
$$24=2(5)+2(7)$$ $$25=5(5)$$ $$26=5+3(7)$$ $$27=4(5)+7$$ $$28=4(7)$$
Then for my induction step, I used the fact that every number n greater than 24 can be written in one of the following forms: $$24+5k$$ $$25+5k$$ $$26+5k$$ $$27+5k$$ $$28+5k$$
I then substitued 24, 25, 26, 27 and 28 by the terms in the base cases to show that in each case, n could be written as a combination of 5 cents and 7 cents.
However, I used the base cases directly in the inductive step, I did not use any induction hypothesis. Is an induction hypothesis really needed in this case, or can we do without one? Thanks!
You used induction implicitly, when asserting that any number greater than $24$ can be written as $x+5k$, with $24\le x\le 28$.
Instead your (strong) induction hypothesis should be: let $n>28$ and assume every integer $k$ with $24\le k<n$ can be obtained with $5$ and $7$ cents.
Then $24\le n-5<n$ so, $n-5=5a+7b$ and therefore $n=5(a+1)+7b$.