If n geometric means are inserted between two quantities ; then the product of n geometric means is the nth power of the single geometric mean between two quantities.

Explanation

Suppose `a` and `b` are two numbers and geometric mean between them are `G_1, G_2,G_3, ldots , G_n`

Then

`a,G_1, G_2,G_3, ldots , G_n,b` are in Geometric progression.

Then the first term =a, and common ratio =r

As `b=a_(n+2)=ar^(n+1)`

`implies r= (b/a)^(1/(n+1))`

Hence, `G_1=ar`

`G_2=ar^2`

...