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
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To prove that the statement is true for every positive integer n using mathematical induction, we need to follow three steps:
1. Base Case: Start by showing that the statement holds for the smallest possible value of n, which is typically 1.
2. Inductive Hypothesis: Assume that the statement is true for some arbitrary positive integer k, which is typically called the "inductive hypothesis."
3. Inductive Step: Prove that if the statement is true for k, it must also be true for k+1.
Let's proceed with the proof:
Base Case:
For n = 1,
n^2 - n + 2 = 1^2 - 1 + 2 = 2
Since 2 is a factor of 2 (2 divided by 2 gives a remainder of 0), the statement holds for n = 1.
Inductive Hypothesis:
Assume that for some k > 1, 2 is a factor of k^2 - k + 2.
Inductive Step:
We need to establish that if the statement holds for k, then it also holds for k+1.
Consider (k+1)^2 - (k+1) + 2:
(k+1)^2 - (k+1) + 2 = k^2 + 2k + 1 - k - 1 + 2 = k^2 + k + 2
Using the inductive hypothesis, we can rewrite k^2 + k + 2 as 2m, where m is some positive integer. This gives:
k^2 + k + 2 = 2m
Adding 2 to both sides of the equation:
k^2 + k + 2 + 2 = 2m + 2
Simplifying, we get:
k^2 + (k+1) + 2 = 2(m + 1)
This shows that (k+1)^2 - (k+1) + 2 can be expressed as 2 times some positive integer (m+1), which means 2 is a factor of (k+1)^2 - (k+1) + 2.
Since we have established the base case and the inductive step, we can conclude by mathematical induction that the statement "2 is a factor of n^2 - n + 2" is true for every positive integer n.