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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. |
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Answer» 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. |
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