Let $\Delta =\left|\begin{array}{ccc}a& a+d& a+2d\\ a+d& a+2d& a\\ a+2d& a& a+d\end{array}\right|$ then

 $c_1\approx c_1+c_2+c_3$
 $\Delta =(3a+3d)\left|\begin{array}{ccc}1& a+d& a+2d\\ 1& a+2d& a\\ 1& a& a+d\end{array}\right|$
 $R_1\approx R_1-R_2,R_2\approx R_2-R_3$
 $=3(a+d)\left|\begin{array}{ccc}0& -d& 2d\\ 0& 2d& -d\\ 1& a& a+d\end{array}\right|$
 $=3(a+d)[d^2-4d^2]=-9d^2(a+d)$
Expanding with respect to C₁