prove by using strong mathmatical induction?

Question

which amount of money can be formed using only two dollar bills and five-dollar bills?

Answer 1

So, which amounts A = 2d + 5f for any non negative integer
values of d and f? If A is odd then A - 5 is even and we can choose

A = 2((A - 5)/2) + 5(1)

This works for A>5.

If A is even then 5 divides one of A,A-2,A-4,A-6,A-8 .
suppose it's A-k where k is one of 0,2,4,6, or 8. Then,

A = 2((k)/2) + 5((A-k)/5)

This works for A>10.

Check for solutions for A +10 or less:

A=1 no solutions
A=2 yes A= 2(1) + 5(0)
A=3 no solutions
A=4 yes A= 2(2) + 5(0)
A=5 yes A= 2(0) + 5(1)
A=6 yes A= 2(3) + 5(0)
A=7 yes A = 2(1) + 5(1)
A=8 yes A = 2(4) + 5(0)
A=9 yes A = 2(2) + 5(1)
A=10 yes A = 2(5) + 5(0) = 2(0) + 5(2)

So all the A's work for A>10 and all the A's except 1 and 3
work for A not greater than ten so the only amounts you can't
get are $1 and $3.

Answer 2

It's so easy it's hard to make the proof formally inductive.

Hmm.

OK. Show explicitly how to do it for the 4 or 5 dollars. But if you can do it for n-2 dollars you can also do it for n dollars.

That's the essence of a proof that you can do it for any whole number amounts greater than or equal to 4.