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Math Help - Harmonic Mean

  1. #1
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    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?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by symmetry View Post
    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
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  3. #3
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    ok

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

    Did you multiply?

    I don't follow here, sorry.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by symmetry View Post
    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
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  5. #5
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    ok

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