If a b and c are in GP and $a^{1 / x} = b^{1 / y} = c^{1 / z}$ , prove that x, y and z are in AP.

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Solution

We have, $a^{1 / x} = b^{1 / y} = c^{1 / z} = k$ [say]

$\Rightarrow a^{1 / x} = k, b^{1 / y} = k$ and $c^{1 / z} = k$

$\Rightarrow a = k^{x}, b = k^{y}$ and $c = k^{z}$

Now, a, b and c are in GP

$\Rightarrow b^{2} = a c$

$∴ (k^{y})^{2} = k^{x} . k^{z}$

$\Rightarrow k^{2 y} = k^{x + z}$

$\Rightarrow 2 y = x + z$

Hence, x, y and z are in AP.

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