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 G_1, G_2, G_{3, …} G_n

Hence a, G_1, G_2, G_{3, …} G_n, b are in GP

⇒ G₁ = ar, G₂ = ar² and so on …

Now, we have n+2 term

⇒ b = ar^n+2-1

⇒ b = arⁿ⁺¹

⇒ … (i)

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

= ar × ar² × ar³ × … arⁿ

= aⁿ × r^{(1+2+3… + n)}

= aⁿ ×

Substituting the value of r from eqn. (i)

= aⁿ ×

= aⁿ ×

= aⁿ ×

=

=

= … (ii)

Single geometric mean between a and b

n^th power of single geometric mean between a and b

Hence Proved