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

In this page explained about 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.

Arithmetic Mean

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

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

So

⇒ q/G = G/p

⇒   G = \sqrt{pq}

Harmonic Mean

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

Here p, H, 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, H are the arithmetic, geometric and harmonic means between p and q, then

A = \frac{p +q}{2 }   ———– ( 1)

 G = \sqrt{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 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 the 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 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 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

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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.

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My self Sivaramakrishna Alluri. Thank you for watching my blog friend

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