Start with 2 numbers, a and b and calculate HM and GM Now you have 2 numbers again, so you can play the game again with the new values Every step brings the results together, one from above, the other from below, sandwiching the value in the limit. That value is called Geometric-Harmonic Mean
This works for all 3 pairs of means (HM-GM, GM-AM, HM-AM). The fun fact I was talking about is about the last combination: playing the game with two "extremal" means, the AM and HM, the value they converge to is GM !!
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1. I kind of did a speed run through high school math, taking essentially 5+ years of math in three years, so it’s likely that I ended up missing/glossing over stuff that people who were learning at a more rational pace did learn, although I think some of my teachers were too intimidated by me to try actually teaching me, much to my detriment.
That's actually one argument for not calling the root mean square a "mean", because a mean should arguably have the property that it is always a number between the largest and smallest value. But the RMS of two negative numbers is positive. (On the other hand, the median would qualify as a mean in this sense, even though it is not a "power mean".)
