1. 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?

2. 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

3. ok

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

Did you multiply?

I don't follow here, sorry.

4. 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

5. ok

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