Actually, this problem reminds me of when I derived the formula on my own in high school. Before there was such a thing as home computers.
Sometimes the way you look at the problem is what provides the solution.
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Maybe a better hint is that since the average = total / items, it follows that the total is the average * items. That should help you derive the formula. You can either derive it directly or subtract the formula for a 1-m series from a 1-n series and reduce the formula for the formula of the sum of a m-n series.