Question

If x,y,z are $pth$ ,$qth$ and $rth$ terms respectively, of an A.P. and also of G.P., then $x^{y-z}{y}^{z-x}{z}^{x-y}$, is equal to

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

Answer 1

The correct option is C $1$

$x,y,z$ are $p^{th},q^{th},r^{th}$ term of $A.P.$ & $G.P.$

$\therefore x=A{R}^{p-1}=B+(p-1)d$

$y=A{R}^{q-1}=B+(q-1)d$

$z=A{R}^{r-1}=B+(r-1)d$

$=x^{y-z},y^{z-x},z^{x-y}$

$\frac{x^y}{x^z},\frac{y^z}{y^x},\frac{z^x}{z^y}$

${\left(\frac{x}{z}\right)}^y,{\left(\frac{y}{x}\right)}^z{\left(\frac{z}{y}\right)}^x$

$R^{y(b-r)}$ $R^{z(q-b)}$ $R^{x(r-q)}$

$\Rightarrow R^{\left[\right(B+(q-1)d(p-r)]+[(B+(r-1\left)d\right(q-p\left)\right]+B+(p-1)d(r-q)]}$

$\Rightarrow R^{\left[B\right(0)+d(pq-pr-q+r+qr-qp-x+p+pr-qr-p+q]}$

$\Rightarrow R^{0+0}=R^0=1$

Option $C$ is correct