Consider a set of \(n\) consecutive terms from an arithmetic progression. Take any 2 terms and replace them by their average. Continue this until only one number remains. Prove that the mean of all end numbers (they need not be distinct) is equal to the median and, for \(n \geq 4\), the mode.

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TopNewestOh wow, wasn't expecting the mean, mode and median to be the same.

I guess that the first step is: "How many different ways are they to do this for \(n\) consecutive terms?"

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