Question

If the $m^{th},n^{th}$ and $p^{th}$ terms of an A.P. and G.P. are equal and are x, y, z then $x^{y-z}{y}^{z-x}{z}^{x-y}$ is equal to-

Answer 1

Then $x=a+(m-1)d$ and $x=b{r}^{m-1}$
$y=a+(n-1)d$ and $y=b{r}^{n-1}$
$z=a+(p-1)d$ and $z=b{r}^{p-1}$
$\therefore x-y=(m-n)d,y-z=(n-p)d,z-x=(p-m)d$
Now $x^{y-z},y^{z-x},z^{x-y}=\left[b{r}^{m-1}{]}^{(n-p)d}\right[b{r}^{n-1}{]}^{(p-m)d}[d{r}^{p-1}{]}^{(m-n)d}$
$=b^{[n-p+p-m+m-n]d}{r}^{\left[\right(m-1\left)\right(n-p)+(n-1\left)\right(p-m)+(m+n\left)\right]d}$
$=b^{0.d}{r}^{0.d}=1$