Question

If the median of a set consisting of \( n \) consecutive odd positive integers is an even integer, the sum of the mean and the range of the set is an odd integer.

• A. Always
• B. Sometimes
• C. Never
  % Correct answer
• D.
• E.

Hint:

When dealing with sets of integers, check both the parity of the median and the behavior of the sum of the range and mean before concluding the relationship.

Answer 1

The Correct Option is B

Consider a set of \( n \) consecutive odd integers. The median of the set will be an odd number if \( n \) is odd and will be an even number if \( n \) is even. For example: - For \( n = 5 \), the set might be \( \{1, 3, 5, 7, 9\} \), with a median of 5 (odd). - The mean is also an odd number in this case. - The range (difference between the largest and smallest integers) will always be odd. Thus, sometimes the sum of the mean and range will be odd, but this is not always true for every set. Hence, the correct answer is "Sometimes".