Question

If $a,b,c$ are in GP and $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{x}}$, then $x,y,z$ are in:

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A AP

Let $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}=k$

$a^{\frac{1}{x}}=k,b^{\frac{1}{y}}=k,c^{\frac{1}{z}}=k$

$a=k^{x}b=k^{y}c=k^z$

$b^2=ac(a,b,c$ are in GP $)$

$k^{2y}=k^x.k^z$

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

$\Rightarrow 2y=x+z$

$y+y=x+z$

$y-x=z-y$

Hence, $x,y,z$ are in A.P