Question

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

much help needed.

Answer 1

1) \(S_1\) is true since \(1^2-1+2\) is divisible by 2.
2) Suppose that \(S_k=k^2-k+2\) is divisible by 2 \(\forall\) positive integers \(k\). Then, we must show that \(S_{k+1}=(k+1)^2-(k+1)+2\) is also divisible by 2.

[Manipulate the last expression to produce two terms, one being \(k^2-k+2\) (which we assume is divisible by 2, and the other having 2 as one of its factors.]

Answer 2

thanks