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

induction is awesome because you can always finish half of the problem without even knowing what the problem is about

how are you stuck on this ?

Answer 2

if we are going to use mathematical induction, we need the basis case, p(k) when n =k case and the p(k+1) when n = k+1 case.

Answer 3

Let \[a(n)=n^2-n+2\]
We want to show that a(n) is even for every interger n.
This is true for n=0 because a(0)=2 which is even.

Now we suppose a(n) is even for a certain n>0 and want to show that a(n+1) is also even.

We have \[a(n+1)=(n+1)^2-(n+1)+2=n(n+1)+2=n^2+n+2=a(n)+2n\]

We see that a(n+1) is the sum of two even numbers : a(n) and 2n.
The result a(n+1) is then an even number.

This ends the induction proof.