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

Let `A^(x)=G^(y)=H^(z)=k`
Then, `A=k^((1)/(x)),G=k^((1)/(y)),H=k^((1)/(z))`
` therefore G^(2)=AH implies (k^((1)/(y)))^(2)=k^((1)/(x))*k^((1)/(z))`
` implies k^((2)/(y))=k^((1)/(x)+(1)/(z))implies (2)/(y)=(1)/(x)+(1)/(z)implies (1)/(x),(1)/(y),(1)/(z)` are in AP.
hence, `x,y,z`are in HP.