Question

Suppose n is an integer. Select all statements below that are true:

A) On^2 + n is always an even integer
B) In^2 + n is always an even integer when n is even
C) In^2 + n is always an even integer when n is odd
D) In^2 + n is never an even integer when n is odd
E) On^2 + n is never an even integer
F) In^2 + n is sometimes an even integer

Answer 1

The expression n2 + n is always even, regardless of whether n is odd or even. Therefore, Statements B, C, and F are true, while Statements A, D, and E are false.

Step-by-step explanation:

The student has been asked to select all statements that are true regarding the expression n2 + n and its parity (whether it is even or odd). We must examine the various cases based on the parity of the integer n.

• Statement A: 0n2 + n is always an even integer - This statement is false because the multiplier of 0 makes the entire first term disappear and the resultant value depends on the parity of n. If n is odd, n2 + n would not be even.
• Statement B: n2 + n is always an even integer when n is even - This statement is true. An even integer squared is even and adding another even integer (n) would result in an even sum.
• Statement C: n2 + n is always an even integer when n is odd - This statement is true as well. An odd number squared is odd, and adding another odd number (n) gives an even sum.
• Statement D: n2 + n is never an even integer when n is odd - This statement is false because, as explained in Statement C, the sum will be even.
• Statement E: 0n2 + n is never an even integer - This statement is false because it disregards cases where n is even or zero.
• Statement F: n2 + n is sometimes an even integer - This statement is true because, as explained in Statements B and C, the expression is always even.