Prove that, if n is an odd number, then n2 + 1 is an even number.

Sure, here is the proof:

Step 1: Assume n is an odd number. By definition, an odd number can be written in the form 2k + 1, where k is an integer.

Step 2: Substitute n = 2k + 1 into the expression n^2 + 1. This gives us (2k + 1)^2 + 1.

Step 3: Expand the square in the expression. This gives us 4k^2 + 4k + 1 + 1.

Step 4: Simplify the expression. This gives us 4k^2 + 4k + 2.

Step 5: Notice that the expression 4k^2 + 4k is clearly even, because it's a multiple of 2.

Step 6: Adding 2 to an even number results in another even number. Therefore, 4k^2 + 4k + 2 is even.

So, if n is an odd number, then n^2 + 1 is an even number.