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    You are at:Home»Pure Math»Algebra»Progressions»Relationship Between Arithmetic, Geometric, Harmonic Mean. AM, GM and HM
    Formulas of Arithmetic Mean, Geometric Mean and Harmonic Mean.  Relation between Arithmetic , Geometric and Harmonic Mean. Also given examples on Arithmetic Progressions (AP),  Geometric Progressions (GP) and Harmonic Progressions.
    Progressions

    Relationship Between Arithmetic, Geometric, Harmonic Mean. AM, GM and HM

    sivaalluriBy sivaalluriOctober 27, 2019Updated:January 16, 2025No Comments3 Mins Read

    This page explains the formulas for Arithmetic Mean, Geometric Mean, and Harmonic Mean and the relationship between them. It also gives examples of Arithmetic Progressions (AP), Geometric Progressions (GP), and Harmonic Progressions.

    Table of Contents

    Toggle
    • Arithmetic Mean
    • Geometric Mean
    • Harmonic Mean
    • Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean ( AM, GM & HM)
    • Examples of AM, GM, and HM

    Arithmetic Mean

    The arithmetic mean “A” of any two quantities of ” p” and ” q”. Then

    Here p, A, and q are in Arithmetic Progressions (AP). Then each being equal to the common difference

    So

    ⇒  q – A = A – p

    ⇒  A = \frac{p + q}{2}

    Geometric Mean

    The Geometric mean “G” of any two quantities of ” p” and ” q”. Then

    Here p, G, and q are in Geometric  Progressions (AP). Then each being to the common ratio

    So

    ⇒ q/G = G/p

    ⇒  G = √pq

    Harmonic Mean

    The Harmonic means “H” of any two quantities of ” p” and ” q”. Then

    Here p, H, and q are in Harmonic  Progressions (AP). Then reciprocals of each being equal to the common difference

    So

    ⇒ \frac{1}{H} - \frac{1}{p} = \frac{1}{q} - \frac{1}{H}

    ⇒   \frac{2}{H} = \frac{1}{p} + \frac{1}{q}

    ⇒  H = \frac{2pq}{p+q}

    Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean ( AM, GM & HM)

    If A, G, and H are the arithmetic, geometric, and harmonic means between p and q, then

     A = \frac{p + q}{2}     ———– ( 1)

    G = √pq    ———— ( 2)

     H = \frac{2pq}{p+q}     ———- ( 3)

    Multiplying equations ( 1) and (3)

    A x H =  \frac{p+q}{2} \times \frac{2pq}{p+q} = pq = \sqrt{G}

    So G2 = AH  ———–  (4)

    Now subtract the equation from (1) to (3)

    A – H =  \frac{p+q}{2} - \sqrt{pq}

    =  \frac{p+q - 2 \sqrt{pq}}{2}

    =  \left[ \frac{\sqrt{p} - \sqrt{q}}{\sqrt{2}} \right]^2   ————-  ( 5 )

    Here the value of ( A – H) is positive if p and q are positive. Therefore, the arithmetic mean of any two possible quantities is greater than their geometric mean.

    Also from equation (4), we see that G is intermediate in value between A and H; and also

    A ≥ G, therefore G ≥ H and A ≥ G ≥ H . ( if p and q are positive)

    Examples of AM, GM, and HM

    Examples – 1: p and q are two numbers whose arithmetic mean is 25 and the Geometric mean is 7. Then find the value of p & q

    Solution: Here AM of p & q is 25. So

    p + q = 50 ———– ( 1)

    GM of p & q is 7. So

    pq = 49   ———– ( 2)

    From the above equations, values of p and q are possible as follows

    p = 49 and q = 1

    Examples – 2: If the A.M and G.M of two numbers are 16 and 8 then find the value of H.M.

    Solution:Here A.M = 16 and G.M = 8

    A x H = G2

    16 x H = 82

    H = 4

    Thanks for reading this article. I hope you liked this “ Relationship Between Arithmetic, Geometric, nk Harmonic Mean  ” article. Give feedback and comments, please.

     

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    Relation between AM, GM and HM

    AP GP and HP Examples Arithmetic Mean Arithmetic progression Geometric Mean harmonic mean harmonic progression Relation Between Am Gm And Hm relation between ap gp and hp
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