The harminic mean of a set of numbers x1, x2,.., xN is defined as:

hm=(1/x1+1/x2+..+1/xN)/N

it is the arithmetic mean of the recipricals of the data values.

It is used for a number of purposes in maths, but one example

is: if you complete a journey of N 1 mile stages with speed v_n

on the n-th stage, then the average speed for the whole journey is:

(1/v_1+1/v-2+..+1/v_N)/N

As a<b, 1/a>1/b, so:QUESTION:

For 0 < a < b, let h be defined by

(1/h) = (1/2)((1/a)+(1/b)).

Show that a < h < b. The number h is called the harmonic mean of a and b.

(1/h) = (1/2)((1/a)+(1/b)) <(1/2)((1/a) + (1/a))=1/a

so h>a.

Also:

(1/h) = (1/2)((1/a)+(1/b)) >(1/2)((1/b) + (1/b))=1/b

so h<b,

hence: a<h<b

RonL