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# If the $i^{th},j^{th}\; and\; k^{th}$ term of an AP and a GP are equal and are $x,y,z$ find the value of $x^{(y-z)}\;.y^{(z-x)}\;.z^{(x-y)}$

$\begin{array}{1 1} xyz \\ \large\frac{x+y+z}{2} \\ 1 \\ (x-y)(y-z)(z-x) \end{array}$

Can you answer this question?

Answer : (c) 1
Explanation :
$x=a+(i-1)d=b\;r^{i-1}$
$y=a+(j-1)d=b\;r^{j-1}$
$z=a+(k-1)d=b\;r^{k-1}$
$x-y=(i-j)\;d$
$y-z=(j-k)\;d$
$z-x=(k-i)\;d$
$x^{y-z}\;.y^{z-x}\;.z^{x-y}= [b\;r^{(i-1)}]^{(j-k)d}\;. [b\;r^{(j-1)}]^{(k-i)d}\;. [b\;r^{(k-1)}]^{(i-j)d}$
$b^{(j-k+k-i+i-j)d}\;.r^{(i-1)(j-k)+(j-1)(k-i)+(k-1)(i-j)d}$
$=b^{0*d}\; r^{0*d}=(br)^0=1.$
answered Jan 2, 2014 by