# If $l,m,n$ are $\;x^{th}\;,y^{th}\;and\;z^{th}$ term of a GP then , $\begin{vmatrix}\log l& x & 1\\\log m &y &1\\\log n &z &1\end{vmatrix}=$

$(a)\;x+y+z\qquad(b)\;xyz\qquad(c)\;xy+yz+xz\qquad(d)\;0$

Explanation : $\;l=ar^{x-1}\quad\;m=ar^{y-1}\quad\;n=ar^{z-1}$
$\Delta=\begin{vmatrix} log a+(x-1)log r &x &1 \\\log a+(y-1)log r &y &1\\\log a+(z-1)log r &z &1 \end{vmatrix}$
$=0.$