Question

If $x,y,z$ are $p^{th},q^{th}$ and $r^{th}$ terms respectively of a $G.P$., then $x^{q-r}\cdot y^{r-p}\cdot z^{p-q}$ is simplified to

• A. $1$
• B. $0$
• C. $xyz$
• D. $Noneofthese$

Answer 1

Answer: (A)

$1$

$x=A{R}^{p-1}$

$y=A{R}^{q-1}$

$z=A{R}^{r-1}$

$x^{q-r}{y}^{r-p}{z}^{p-q}$

$={\left(A{R}^{p-1}\right)}^{q-r}{\left(A{R}^{q-1}\right)}^{r-p}{\left(A{R}^{r-1}\right)}^{p-q}$

$=A^{\left(q-r+r-p+p-q\right)}{R}^{\left[\left(p-1\right)\left(q-r\right)+\left(q-1\right)\left(r-p\right)+\left(r-1\right)\left(p-q\right)\right]}$

$=A^0{R}^{\left[pq-pr-q+r+qr-pq-r+p+pr-qr-p+q\right]}$

$=A^0{R}^0$

$=1$