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
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.
To prove that the statement "2 is a factor of n^2 - n + 2" is true for every positive integer n using mathematical induction, we need to do the following:
Step 1: Base Case
We begin by checking if the statement is true for the smallest value of n, which is usually n = 1 or sometimes n = 0. In this case, let's check if 2 is a factor of 1^2 - 1 + 2.
When n = 1:
n^2 - n + 2 = 1^2 - 1 + 2 = 2.
Since 2 is indeed divisible by 2, the base case holds true.
Step 2: Inductive Hypothesis
Assume that the statement is true for some positive integer k, where k is an arbitrary positive integer. So, assume that 2 is a factor of k^2 - k + 2.
Step 3: Inductive Step
Now, we need to show that the statement is true for the next positive integer, which is k+1. In other words, we need to show that 2 is a factor of (k+1)^2 - (k+1) + 2.
(k+1)^2 - (k+1) + 2
= k^2 + 2k + 1 - k - 1 + 2
= k^2 + k + 2k + 2
Rearranging the terms:
= k^2 - k + 2k + 2
= (k^2 - k + 2) + 2k
= (divisible by 2) + 2k
Since we assumed that k^2 - k + 2 is divisible by 2 (inductive hypothesis), we can rewrite the expression as:
= (2m) + 2k (where m is some positive integer)
Factoring out 2:
= 2(m + k)
Since (m + k) is also a positive integer, we can conclude that (k+1)^2 - (k+1) + 2 is divisible by 2.
Step 4: Conclusion
By completing the base case, the inductive hypothesis, and the inductive step, we have shown that if the statement is true for a positive integer k, then it is also true for the next positive integer k+1. Therefore, by the principle of mathematical induction, the statement "2 is a factor of n^2 - n + 2" is true for every positive integer n.