Let `a_(1), a_(2)`...be positive real numbers in geometric progression. For n, if `A_(n), G_(n), H_(n)` are respectively the arithmetic mean, geometric mean and harmonic mean of `a_(1), a_(2),..., a_(n)`. Then, find an expression for the geometric mean of `G_(1), G_(2),...,G_(n)` in terms of `A_(1), A_(2),...,A_(n), H_(1), H_(2),..., H_(n)`

Explanation

To solve the problem, we need to find an expression for the geometric mean of the geometric means \( G_1, G_2, \ldots, G_n \) in terms of the arithmetic means \( A_1, A_2, \ldots, A_n \) and harmonic means \( H_1, H_2, \ldots, H_n \) of the terms in a geometric progression.

### Step-by-Step Solution:

1. **Understanding the Means**:
- Given that \( a_1, a_2, \ldots, a_n \) are in geometric progression, we can express them as:
\[
a_1 = A, \quad a_2 = AR, \quad a_3 = AR^2, \ldots, \quad a_n = AR^{n-1}
...