Question

If $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}$ and $a,b,c$ are in G.P., then which of the following are true?

• A. $x,y,z$ will be in A.P.
• B. $x,y,z$ will be in G.P.
• C. $x,y,z$ will be in H.P.
• D. None of the above

Answer 1

The correct option is A

$x,y,z$ will be in A.P.

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

Since $a,b,c$ are in G.P., $b^2=ac.$
$\therefore k^{2y}=k^x.k^z=k^{x+z}$
$\therefore 2y=x+z$
$\Rightarrow x,y,z$ are in A.P.