For any integer n, the expression n^2 + n + 7 is an odd integer. — GMAT

For any integer $n$ , the expression $n^2 + n + 7$ is an odd integer.

Answer: Answer

The statement is true because

n(n + 1)

is the product of two consecutive integers, which is always even. Adding the odd constant

7

to an even integer yields an odd integer for all integer values of

n

The statement is true because

n^2 + n = n(n + 1)

represents the product of two consecutive integers and is therefore always even. Adding

7

(an odd integer) to an even integer always yields an odd integer regardless of whether

n

is positive, negative, or zero.

Factor the algebraic terms

n^2 + n

n^2 + n + 7 = n(n + 1) + 7

Factoring out

n

reveals the product of two consecutive integers.

Determine the parity of the product of two consecutive integers

n(n + 1)

n(n + 1)

is always even.

Of any two consecutive integers, exactly one is even. The product of an even integer and any integer is always even.

Determine the overall parity of the expression by adding

7

\text{even} + 7 = \text{odd}

Adding an odd integer to an even integer results in an odd integer.

The product of two consecutive integers is always even. Adding an odd integer to an even integer results in an odd integer.

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