Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

The correct option is C H.P.

$a,b,c$ in $GP\Rightarrow b=ar,c=a{r}^2$

$\therefore a^x=(ar{)}^y=(a{r}^2{)}^2$

$a^{x-y}=r^y;a^{x-z}=r^{2z}$

$\Rightarrow \frac{x-y}{x-2}=\frac{y}{2z}$

$\Rightarrow 2xz-2yz=xy-yz$

$\Rightarrow \frac{2xz}{xyz}=\frac{xy+yz}{xyz}$

$\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$\therefore x,y,z$ are in HP.