Question

Prove that if n is odd then 3n+5 is even.

A. By induction
B. By contradiction
C. By direct proof
D. By counterexample

Answer 1

Using a direct proof, if n is odd then 3n+5 is even because multiplying an odd number by 3 and then adding an odd number results in an even number.

Step-by-step explanation:

To prove that if n is odd then 3n+5 is even, we would use a direct proof. An odd number is always one more than an even number, which can be represented as 2k+1, where k is any integer. Therefore, if n is odd, n = 2k+1 for some integer k.

Now, multiplying the odd number by 3, we get 3(2k+1) = 6k+3, which simplifies to 6k+2+1. Here, 6k+2 is clearly even because it's 2 multiplied by an integer (3k+1), and adding 1 to it results in an odd number. But when we add 5 (which is odd) to this odd number, we have an even result because adding two odd numbers always yields an even number. Hence, 3n+5 = (6k+2+1)+5 = 6k+2+(1+5), and since 1+5=6, which is even, we have 6k+2+6, which is an even number plus another even number, confirming that 3n+5 is even.