Relationship Between Arithmetic, Geometric, Harmonic Mean

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.

Contents

1 Arithmetic Mean
2 Geometric Mean
3 Harmonic Mean
4 Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean ( AM , GM & HM)
5 Examples of AM, GM and HM

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

⇒ 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

⇒ 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

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

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

⇒ $H = \frac{2pq}{p+q}$

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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