from https://www.quora.com/Im-thinking-of...tany?srid=oNrJ

The three numbers are 2,3,4 and the “average” you are thinking of is a generalized mean:

Mp= [(2^{p}+3^{p}+4^{p})/3] ^ (1/p) =π

.

The arithmetic mean isM1=[2+3+4] / 3=3

but you wouldneverthink of such aboringaverage.

The geometric mean isM0=limq→0Mq= [2*3*4]^(1/3) ≈ 2.8845

.

The quadratic mean isM2 = [(4+9+16)/3] ^ (1/2) ≈ 3.1091

.

The cubic mean isM3 = [(8+27+64)/3] ^ (1/3) ≈ 3.2075

.

It turns out that the generalized mean is a continuous monotonic increasing function ofp

so, by the intermediate value theorem, there is some 2<p<3 such thatMp=π.

Clearlytheaverage about which you were thinking ⌣¨

Of course the boring old arithmetic mean of any finite set of Rational numbers is Rational. It is therefore neverπ

because that value is known to be Irrational, in fact Transcendental.

More evidence that you weren’t thinking of this average or, indeed, any generalized mean with an Integer value forp.

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