If a, b, c are in a geometric progression and | vedclass

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(B) Given that $a, b, c$ are in a geometric progression,we have $b^2 = ac$ $(1)$.Let $a^x = b^y = c^z = k$.Then $a = k^{1/x}, b = k^{1/y}, c = k^{1/z}

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Harmonic Progression

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(B) Given that $a, b, c$ are in a geometric progression,we have $b^2 = ac$ $(1)$.Let $a^x = b^y = c^z = k$.Then $a = k^{1/x}, b = k^{1/y}, c = k^{1/z}$.Substituting these values into equation $(1)$:$(k^{1/y})^2 = k^{1/x} \cdot k^{1/z}$$k^{2/y} = k^{1/x + 1/z}$Equating the exponents: $\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$.This implies that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an arithmetic progression.Therefore,$x, y, z$ are in a harmonic progression.

If $a, b, c$ are in a geometric progression and $a^x = b^y = c^z$,then in which progression are $x, y, z$?

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