If $a, b, c$ are in geometric progression and $a^x = b^y = c^z$, then which of the following relations holds true between $x, y, z$?

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KollegeApply · Accepted Answer

The correct option is (A): $y = rac{2xz}{x+z}$ (or $ rac{1}{x} + rac{1}{z} = rac{2}{y}$, meaning $x, y, z$ are in Harmonic Progression).Let $a^x = b^y = c^z = k \implies a = k^{1/x}, b = k^{1/y}, c = k^{1/z}$.Since $a, b, c$ are in GP, $b^2 = ac \implies (k^{1/y})^2 = k^{1/x} \cdot k^{1/z} \implies k^{2/y} = k^{1/x + 1/z}$.Thus $ rac{2}{y} = rac{1}{x} + rac{1}{z} \implies y = rac{2xz}{x+z}$.

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