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 $H.P$.