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This problem trains for: SAT-I.

The average of an even number of consecutive odd numbers can be equal to:

If a set of numbers is formed of numbers that are equally spaced (arithmetic sequence), then the average is equal to the median value.

If there is an even number of numbers in the sequence, the median is the average of the two middle values. Since these values are consecutive odd numbers, they could be written as:

and their average is always an even number:

which makes the first answer choice true.

The median of an even number of even numbers is always odd:

The median of an odd number of odd numbers is odd.

A power of 3 is always odd.