Given that $a,b,c$  are in AP.Then determinant $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$  in its simplest form is equal to

$\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$ 

Operate $C_1\to C_1-C_{ 2}$

   =     $\left|\begin{array}{ccc}-1& x+2& x+a\\ -1& x+3& x+b\\ -1& x+4& x+c\end{array}\right|$

       $=(-1)\left|\begin{array}{ccc}1& x+2& x+a\\ 1& x+3& x+b\\ 1& x+4& x+c\end{array}\right|$

Operate $R_2\to R_2-R_1,R_3\to R_3-R_1$

       $=(-1)\left|\begin{array}{ccc}1& x+2& x+a\\ 0& 1& b-a\\ 0& 2& c-a\end{array}\right|$

      $=\left(-1\right)\left[c+a-2b\right]=0$   

                 $\left[\because a,b,careinAP\Rightarrow 2b+a+c\right]$