Question

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work.
2 is a factor of n2 -n+2

Answer 1

if n is even, so is n^2, so the sum is even
if n is odd, so is n^2, and since the sum of odd+odd is even, so is the whole thing.

But that's just logic. To prove it, take

S1: 2 is a factor of 1^1+1-2 = 0
So, assume Sn.
Sn+1: 2 is a factor of (n+1)^2 - (n+1) + 2
= n^2+2n+1 - n - 1 + 2
= n^2-n+2 + 2n+1-1
= Sn + 2n
So, Sn+1 is also true.

So,
S1
Sn ==> Sn+1
Hence, true for all n.