geometric mean

Geometric mean:

If a, b, c are in G.P . Then middle term of this G.P is known as geometric mean of a G.P.

let b be geometric mean in a, b, c .

     b2 =  a.c

1. Single geometric mean of n positive numbers :

Let G be the geometric mean of n positive numbers—

a1 , a2 , a3 , ……… , an.

G = (a1.a2.a3…..an )1/n

For n = 2

In a , G , b are in G.P

   G = ab

2. n Geometric mean between two numbers :

Let G1 , G2 , G3 , ……. , Gn be the Geometric mean between a and b.

Then ,

a , G1 , G2 , G3 , ………. , Gn , b will be in G.P.

here ,

a =  1st term

let r = common ratio

b = ( n + 2) th term of an G.P

b =   a.rn+2-1

r =  ( b / a )1/n+1

G1 = ar                  =     a. ( b / a )1/n+1

G2 = a.r2              =     a.(b / a )2/n+1

:

Gn =  a.rn           =    a.( b / a )n/n+1

 

Gn = a.( b / a )n/n+1