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!