Problem:
An iterative average of the numbers , and is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
Answer Choices:
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Solution:
If the numbers are arranged in the order , then the iterative average is
The largest value is obtained by letting or , and the smallest value is obtained by letting or . In the former case the iterative average is , and in the latter case the iterative average is , so the desired difference is
The problems on this page are the property of the MAA's American Mathematics Competitions