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

Question

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

Infinity Learn · Accepted Answer

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⋅$$R(x$$−$$1)(y$$−$$z)+(y$$−$$1)(z$$−$$x)+(z$$−$$1)=log$$⁡$$A0$$×$$R0=log$$⁡$$1=0$$.

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

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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=

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