Answer :

Let us suppose a and b are two numbers.


Let us say G is the Geometric mean of a and b.


a, G and b must be in Geometric Progression or GP.


This means, common ratio = G/a = b/G


Or, G2 = ab


Or, Gn = n(ab) ............ (1)


Now, let us say G1 , G2 , G3 ,.......Gn are n geomteric means between a and b.


Which means that


a , G1 , G2 , G3 ...... Gn , b form a G.P.


Note that the above GP has n+2 terms and the first term is a and last term is b, which


is also the (n+2)th term


Hence, b = arn+2-1`


where a is the first term.


So,


b = arn+1


...(2)


Now the product of GP becomes


Product = G1G2G3......Gn


= (ar)(ar2)(ar3)..(arn)


= an.r(1+2+3…+n)


=


Putting the value of r from equation 2 , we get


=


= .


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