Answer : To prove: Product of n geometric means between a and b is equal to the nth power of the single GM between a and b.

Formula used:(i) Geometric mean between

(ii) Sum of n terms of A.P.

Let the n geometric means between and b be G1, G2, G3, … Gn Hence a, G1, G2, G3, … Gn, b are in GP

⇒ G1 = ar, G2 = ar² and so on … Now, we have n+2 term

⇒ b = arn+2-1

⇒ b = arn+1

The product of n geometric means is G1× G2× G3× … Gn

= ar × ar2 × ar3 × …

= an × r(1+2+3… + n)

Substituting the value of r from eqn.

Single geometric mean between a and b

nth power of single geometric mean between a and b Hence Proved