Steve Witham <sw@tiac.net> wrote:

"Keith F. Lynch" <kfl@KeithLynch.net>

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-1: Harmonic mean

Kewl, now I think I can average two resistors.

That depends on whether the resistors are in series or parallel. And on whether you're averaging their resistances or their conductances. Harmonic means show up in lots of different contexts. This morning, I happened to read a blog comment dated just three days ago, which said, in part: Or there was this think PISA question "We traveled to a city at a speed of 20km/h and traveled back at 30 km/h, what was our average speed for the whole journey?" And very few people get it right and not because they are stupid but because they do not use pen and paper to work it out. Why? If you just go "assume they are 100km away" or "assume the trip there was one hour" and work it out on paper there is no way you get it wrong. Seriously. I don't know why people don't do that. http://esr.ibiblio.org/?p=8558 The correct answer is of course the harmonic mean of 20 and 30, which is 24. I solved it my head with no pen, paper, assumed distance, or assumed duration.

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0: Geometric mean

I got as far as wondering whether this was something like what happens when integrating x^(-1)... instead of getting (x^0) / 0, and then went to wikipedia... which treats it like a convention:

Do these means change smoothly as x crosses zero?

Yes. For what it's worth, my program which generated my first table gave the correct answer, 0.000000, for 4,2, 9,3, 16,4, etc., without treating them as special cases. (Okay, some of them gave -0.000000, and I "cheated" by manually deleting the minus sign before posting.) Note that I carefully defined it as (a^x + b^x) / 2 = c^x, not as c = the xth root of ((a^x + b^x) / 2). Trying to take the zeroth root of a number is problematic. Raising a number to the zeroth power is not. My program sets b to 1, then for each pair of positive integers c<a<1000 tries various values of x, quickly converging on the solution with a binary search. It takes less than a minute to fill the table, which has about half a million entries. Related problem: What if everything is an integer? I've found cases where there are positive integers a<b such that there are three solutions to (a^x + b^x) / 2 = c^x in which c and x are both integers. I haven't found any where there are four such solutions. Can you find one or prove there aren't any? Another related problem: Find an infinite integer sequence in which no two terms have an integer power mean in the sequence. For instance A005836 is an infinite integer sequence in which no two terms have an arithmetic mean in the sequence. I want the same thing but also for geometric means, harmonic means, quadratic means, etc., all at the same time.