What\'s wrong with the following proof by induction? Any amount of change greate

Question

What's wrong with the following proof by induction? Any amount of change greater than or equal to twenty cents can be gotten with a combination of five-cent and seven-cent coins. The proof is by induction on the amount of change 1. For twenty cents use four five-cent coins. 2. Let n > 20 be the amount of change. Assume that n 7 5y for some non-negative integers r and y. For any n > 20, either x > 1, or y > 3. If x > 1, then since 3(5) - 2(7) n + 1-5(y + 3) + 7(z-2). If y > 3, then since 3(7)-4(5)-1, n + 1-7(z + 3) + 5(y-1). In either case, we showed that n +1-7u +5v where u and v are non-negative integers.

Explanation / Answer

We have also to prove that if results holds for n=20, then it also holds for n+1=21. We hypothesis is for n>20. In this proof we are missing the proof for n=21.

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