If `A^(x)=G^(y)=H^(z)`, where `A,G,H` are AM,GM and HM between two given quantities, then prove that `x,y,z` are in HP.

Explanation

To prove that \( x, y, z \) are in Harmonic Progression (HP) given that \( A^x = G^y = H^z \), where \( A, G, H \) are the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Set up the equations**:
Given that \( A^x = G^y = H^z \), we can denote this common value as \( k \). Therefore, we have:
\[
A^x = k, \quad G^y = k, \quad H^z = k
...