Question

Use math induction to show that (n2 - n + 2) is divisible by 2 for all natural numbers n.

Answer 1

Here's the process of induction.
When n=1, 1^2-1+2=2. 2 is divided by 2.
When n=2, 2^2-2+2=4. 4 is divided by 2.
When n=3, 3^2-3+2=8. 8 is divided by 2.
Suppose when n=k, k^2-k+2 is divided by 2.
Let n=k+1. Then
(k+1)^2-(k+1)+2=k^2+2k+1-k-1+2
=k^2+k+2=k(k+1)+2
Note one of k and k+1 must be even because they are consecutive natural numbers. Then k(k+1)+2 will be divisible by 2. Consequently, n^2-n+2 is divided by 2 for all natural numbers n.

Answer 2

Marly, is that 1^(2-1+2) or is it 1^2-1+2?