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

Contents

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

⇒

### 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

⇒

### 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

⇒

⇒

⇒

### 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

———– ( 1)

———— ( 2)

———- ( 3)

Multiplying equations ( 1) and (3)

A x H =

So G^{2} = AH ———– (4)

Now subtract equation from (1) to (3)

A – H =

=

= ————- ( 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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