Use the properties of even and odd integers that are listed in Example 4.2.3 to

Question

Use the properties of even and odd integers that are listed in Example 4.2.3 to determine whether the following statement is true or false. Indicate which properties you use to justify your reasoning. “If x is any odd integer, then x 2 + x is even.”

Explanation / Answer

if x is odd, then x is in the form of 2k+1 where k is any integer. => x^2 + x => (2k + 1)^2 + (2k+1) => 4k^2 + 1 + 4k + (2k+1) => 4k^2 + 6k + 2 => 2(2k^2 + 3k + 1) This is in the form of 2n (n = 2k^2 + 3k + 1) since this is in the form of 2n, it is always even. so, If x is any odd integer, then x^2 + x is even. Node: ^ is a power symbol

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