Final answer:
To prove that if 3n + 2 is even, then n is even, we can use a proof by contraposition or a proof by contradiction. In a proof by contraposition, we assume that n is odd and show that 3n + 2 is odd. In a proof by contradiction, we assume that 3n + 2 is even and n is odd, and show that this assumption leads to a contradiction.
Step-by-step explanation:
Proof by contraposition:
To prove that if 3n + 2 is even, then n is even, we can use a proof by contraposition. The contrapositive of this statement is: If n is odd, then 3n + 2 is odd. To prove this, we can assume that n is odd and show that 3n + 2 is indeed odd.
Suppose n is an odd integer. Then, we can write n as n = 2k + 1, where k is an integer. Substituting this into the expression 3n + 2, we get 3(2k + 1) + 2 = 6k + 5. This expression can be written as 2(3k + 2) + 1, which is an odd number. Therefore, if n is odd, then 3n + 2 is odd, proving the contrapositive. Since the contrapositive is true, the original statement is also true, and we can conclude that if 3n + 2 is even, then n is even.
Proof by contradiction:
To prove that if 3n + 2 is even, then n is even, we can use a proof by contradiction. Assume that 3n + 2 is even and n is odd. We will show that this assumption leads to a contradiction.
Suppose n is an odd integer. Then, we can write n as n = 2k + 1, where k is an integer. Substituting this into the expression 3n + 2, we get 3(2k + 1) + 2 = 6k + 5. Since 6k is always even for any integer k, adding an odd number (5) will result in an odd number. Therefore, if n is odd, then 3n + 2 is odd, which contradicts our assumption that 3n + 2 is even. Thus, our initial assumption was incorrect, and we can conclude that if 3n + 2 is even, then n is even.