This discussion reminds me of something I found strange in math. I'd always heard of the "mean" as the "average", or sum of the elements divided by the number of elements. This is also known as the "arithmetic mean."  The arithmetic mean is also the middle number in an arithmetic sequence (where each successive element is the previous element plus a constant). For example, in the sequence: 1 3 5 7 9 where each element is the previous one plus 2, the arithmetic mean is the center element, or 5. This also fits the "average" definition of arithmetic mean since the sum of the entries is 25 and there are 5 elements (25/5=5).

Then there's the "geometric mean." If we have a geometric sequence (where each element is the previous element times a constant), the geometric mean is the middle element (just as the arithmetic mean, above, but this is for a geometric sequence). For example, if the constant is 3, a series might be 1, 3, 9, 27, 81. The geometric mean of 3 and 27 would be 9. This is also the square root of the "end elements" of the sequence (3 and 27). Trying it further, the geometric mean of 1, 3, 9, 27, 81 would be the middle element of 9. The geometric mean of 9, 27, 81 would be 27. Checking the "square root of end elements" method, we have sqrt(1*81)=9 for the first example, and sqrt(9*81)=sqrt(729)= 27, again, the middle element.

So, what's a common use of the geometric mean? It's the resonant frequency of a tuned circuit when you are given the upper and lower 3dB frequencies (FL and Fh). Fc=sqrt(FL*Fh) .

I always wondered why these two means had the names they did. When I later learned their mean-ing in sequences, I thought it was interesting.

Harold

FCC Rules Online at http://www.hallikainen.com


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