Question

Suppose that m and n are consecutive integers. Use a direct proof to show that their sum m+n is an odd integer.

Answer 1

Let's assume that m and n are consecutive integers. Without loss of generality, let's assume that m is the smaller integer and n is the larger integer, so n = m + 1.

We want to prove that m + n is an odd integer. To do this, we can show that m + n can be expressed as 2k + 1 for some integer k.

m + n = m + (m + 1) = 2m + 1

Let k = m. Then 2m + 1 = 2k + 1, which is an odd integer.

Therefore, we have shown that if m and n are consecutive integers, then their sum m + n is an odd integer.