Question

Choose all of the following expressions which are even for all integer values of n.

2n + 12
3n + 6
2n + 7
4n - 10
n + 2

Answer 1

The expressions 2n + 12 and 4n - 10 are even for all integer values of n, because the sum or the difference of two even numbers is always even.

Step-by-step explanation:

To determine which expressions are even for all integer values of n, we must check if the expressions yield an even number for any integer n. An even number is any number that can be divided by 2 without leaving a remainder. Let’s evaluate the given expressions:

• 2n + 12: This expression is always even because 2n represents any even number (since n is an integer and 2 times any integer is even), and 12 is even. Hence, the sum of two even numbers is always even.
• 3n + 6: This expression can be even or odd, depending on the value of n. If n is even, the expression is even, but if n is odd, the expression is odd since 3 times an odd number is odd, and the sum of an odd and an even number is odd.
• 2n + 7: Similar to the previous, this expression can also be even or odd. 2n is even, but since 7 is odd, the sum will be odd, regardless of the value of n.
• 4n - 10: This expression is always even because 4n represents an even number (4 times any integer is even), and 10 is even. So, the difference of two even numbers is always even.
• n + 2: This expression can be even or odd similar to 3n + 6, depending on whether n is even or odd.

Hence, the expressions 2n + 12 and 4n - 10 are even for all integer values of n.