Question

If $a^x = b^y = c^z$ and $a, b, c $ are in G.P. then $ x, y, z $ are in

• A. A.P
• B. G.P
• C. H.P
• D. none of these.

Answer 1

Answer: (C)

$a^{x} =b^{y} = c^{z} = K$ (say)
$ \Rightarrow a= K^{\frac{1}{x}}, b= K^{\frac{1}{y}}, c= K^{\frac{1}{z}}$.
$ \Rightarrow $ Since $a, b, c$ are in $G.P$.
$ \Rightarrow b^{2} =ac$
$ \Rightarrow K^{\frac{2}{y}} = K^{\frac{1}{x}} \cdot K^{\frac{1}{z}} = K^{\frac{1}{x}+\frac{1}{z}}$
$ \Rightarrow \frac{2}{y} = \frac{1}{x}+\frac{1}{z} $
$ \Rightarrow \frac{1}{x} , \frac{1}{y}, \frac{1}{z}$ are in $A.P$.
$ \Rightarrow x, y, z$ are in $H.P$.