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Question Number 100570 by bshahid010@gmail.com last updated on 27/Jun/20

Commented by Dwaipayan Shikari last updated on 27/Jun/20

$$a^x=b^y=c^z=d^w=k(k\ne 0)$$$$a=k^{\frac{1}{x}}b=k^{\frac{1}{y}}c=k^{\frac{1}{z}}d=k^{\frac{1}{w}}$$$$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}(AstheyareinG.P)$$$$ad=bcso{k}^{\frac{1}{x}+\frac{1}{w}}=k^{\frac{1}{y}+\frac{1}{z}}\Rightarrow \frac{1}{x}+\frac{1}{w}=\frac{1}{y}+\frac{1}{z}$$$$orac=b^2{k}^{\frac{1}{x}+\frac{1}{z}}=k^{\frac{2}{y}}\Rightarrow \frac{1}{x}+\frac{1}{z}=\frac{2}{y}$$$$or{c}^2=bd{k}^{\frac{1}{y}+\frac{1}{w}}=k^{\frac{2}{z}}\Rightarrow \frac{1}{y}+\frac{1}{w}=\frac{2}{z}$$$$TheyareinH.P$$

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