If a, b, c are respectively the xth, yth and zth terms of a  G.P., then (y−z)log⁡a+(z−x)log⁡b+(x−y)log⁡c=

Let A be the first term and R, the common ratio of G.P.Then  a=ax=ARx−1,b=av=ARy−1and c=az=ARz−1∴(y−z)log⁡a+(z−x)log⁡b+(x−y)log⁡c=log⁡ay−z+log⁡bz−x+log⁡cx−y=log⁡ay−z⋅bz−x⋅cx−y=log⁡ARx−1y−z×ARy−1z−x⋅ARz−1x−y=log⁡A(x−y)y−z+z−x+x−y⋅R(x−1)(y−z)+(y−1)(z−x)+(z−1)=log⁡A0×R0=log⁡1=0.