Question

If $x,y$ and $z$ are $p^{th},q^{th}$ and $r^{th}$ terms respectively of an A.P. and also of a G.P., then $x^{y-z}{y}^{z-x}{z}^{x-y}$ is equal to

• A. $xyz$
• B. $0$
• C. $1$
• D. None of these

Answer 1

The correct option is C $1$

Given that $x,y,z$ are the $p^{th},q^{th}$ and $r^{th}$ terms of an A.P.
$\therefore x=A+(p-1)D$
$y=A+(q-1)D$
$z=A+(r-1)D$
$\Rightarrow x-y=(p-q)D$
$y-z=(q-r)D$
$z-x=(r-p)D$
where $A$ is the first term and $D$ is the common difference.

Also, $x,y,z$ are the $p^{th},q^{th}$ and $r^{th}$ terms of a G.P.
$\therefore x=a{R}^{p-1},y=a{R}^{q-1},z=a{R}^{r-1}$
$\therefore x^{y-z}{y}^{z-x}{z}^{x-y}=\left(a{R}^{p-1}{)}^{y-z}\right(a{R}^{q-1}{)}^{z-x}(a{R}^{r-1}{)}^{x-y}=(aR{)}^{\circ }=1$