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

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

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

⇒ 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

⇒ 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

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

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

⇒ $H = \frac{2pq}{p+q}$

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

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