Given that $a^{x} = b^{y} = c^{z} = d^{u}$ (where a, b,c, d > 0) and a, b, c, d are in G.P. Then

x, y, z, u are in H.P. for infinite possibilities of a, b, c and d.

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x, y, z, u may be in A.P. for some value of a, b, c and d.

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x, y, z, u may be in A.G.P. for some value of a, b, c and d.

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x, y, z, u can’t be in A.P. for any value of a, b, c and d.

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Solution

The correct options are
A x, y, z, u are in H.P. for infinite possibilities of a, b, c and d.
B x, y, z, u may be in A.P. for some value of a, b, c and d.
C x, y, z, u may be in A.G.P. for some value of a, b, c and d.
$\Rightarrow a^{x} = a^{y} . r^{y} = a^{z} . r^{2 z} = a^{u} . r^{3 z} = K \Rightarrow x = l o g_{a} K y = \frac{l o g_{a} K}{1 + l o g_{a} r} z = \frac{l o g_{a} K}{1 + 2 l o g_{a} r} u = \frac{l o g_{a} K}{1 + 3 l o g_{a} r}$
$\Rightarrow$ x, y, z, u are in H.P.
$\Rightarrow$ x, y, z, u may be in AP./A.G.P.
for a = b = c = d = 1.

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