Thread: Harmonic Mean

What exactly is meant by the term HARMONIC MEAN?

I think of the word harmonic in terms of music but not in terms of mathematics.


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.

Also, is there another purpose for the harmonic mean other than just proving the above?

Originally Posted by symmetry

What exactly is meant by the term HARMONIC MEAN?

I think of the word harmonic in terms of music but not in terms of mathematics.

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

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.

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

(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

Did you add the fractions to find 1/a and 1/b?

Did you multiply?

I don't follow here, sorry.

Originally Posted by symmetry

Did you add the fractions to find 1/a and 1/b?

Did you multiply?

I don't follow here, sorry.

No I replaced 1/a by 1/b (and vice verse) when I needed to get two terms
each the same. That is how the inequalities arrise, since 0<a<b implies
1/a>1/b.

The idea is if 1/a>1/b, then 1/b+1/b<1/a+1/b<1/a+1/a

RonL

I like the harmonic mean idea when used in areas other than proving.