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