If $a^{x} = b^{y} = c^{z}$ and a , b , c be in the G . P . then prove that x , y , z are in H . P

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Solution

$b e c a u s e$ a , b , c are in G . P
$∴ b^{2} = a c$
Also $a^{x} = b^{y} = c^{z} = k s a y$
$∴ a = k^{1 / x}, b = k^{1 / y}, c = k^{1 / z}$ Put the value of a , b , c in (1)
$k^{2 / y}, = k^{a / x} \cdot k^{1 / z}, = k^{1 / x + 1 / z}$
$∴ \frac{2}{y} = \frac{1}{x} + \frac{1}{z}$
Hence $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in A . P
or x , y , z , in H . P

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