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Geometric Mean

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Geometric Mean - Lesson Summary

The geometric mean (GM) between two numbers is a number, which, when placed between them, forms with them a geometric progression.
 
Formula to calculate the geometric mean
 
Let m be the geometric mean between two numbers a and b, then a, m, b are in GP.


The common ratio is m/a or b/m, such that m/a =b/m ⇒ m 2 = ab (m = ab ).


The Geometric Mean (GM) between a and b is ab
 
Ex:
Let a = 4 and b = 16

Then ab   =  4 x 16   =  64  = 8

4, 8, 16 form a GP.
 
In general, many numbers can be inserted between any two given numbers.
 
Let us find n numbers G 1, G 2, G 3,....,G n between two numbers a and b.
 
Then the sequence a, G 1, G 2, G 3,....,G n ,b forms a GP having (n+2) terms.

Let a 1 = a, a 2 = G 1, a 3 = G 2 , .....,  a n+1= G n, a n+2= b.

Now, a n+2 = b

⇒ ar n+2-1 = b, where r is the common ratio of this GP

    ar (n+1) = b

    r (n+1) = b/a
 
    r = b a 1 n+1
 
Then  G 1 = t 2 = ar

      = a x b a 1 n+1    = a b a 1 n+1    

         G 2 = t 3 = ar 2

      = a x b a2 n+1    = a b a2 n+1    

      G 3 = t 4 = ar 3 = a x b a3 n+1    = a b a3 n+1     ……….. and so on.

Likewise, we have G n = t n+1 = ar n = a x b an n+1    = a b an n+1   
 
Relationship between AM and GM

The arithmetic mean between two numbers is their average.


The arithmetic mean (AM) between two numbers is a number, which, when placed between them, forms with them an arithmetic progression.
 
Let A and G be the AM and GM of two positive real numbers a and b, respectively.

Then A = a + b 2   and =  ab .

Now, A - G =  a + b 2   –  ab

                 =  a + b - 2 ab 2

                = a 2 + b 2 - 2 a b 2   

                =  a - b 2 2 ≥ 0

∴  A – G ≥ 0

⇒ A ≥ G

The AM between two positive real numbers is always greater than or equal to their GM.

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