Question

If we wanted to show that if n is an integer and 3n + 2 is even, then n is even by using a proof by contradiction, what should our proof attempt to show?

Answer 1

To use proof by contradiction to show if n is an integer and 3n + 2 is even then n is even, we assume n is odd and show this leads to a contradiction, thus proving n must be even.

Step-by-step explanation:

If we wanted to show that if n is an integer and 3n + 2 is even, then n is even by using a proof by contradiction, our proof should begin by assuming the opposite of what we want to prove; namely, that n is odd. To do this, let us represent n as 2k + 1, where k is an integer (since all odd numbers can be expressed in this form). If we plug this into the equation 3n + 2, we get 3(2k + 1) + 2, which simplifies to 6k + 3 + 2 = 6k + 5. Since 6k is clearly even (as 6 is even), adding 5 yields an odd number, contradicting the assumption that 3n + 2 is even. This contradiction indicates that our original assumption that n is odd is false, hence n must be even.

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