Given that a,b,c  are in AP.Then determinant x+1x+2x+ax+2x+3x+bx+3x+4x+c  in its simplest form is equal to

x+1x+2x+ax+2x+3x+bx+3x+4x+c Operate C1→C1−C 2   =     -1x+2x+a-1x+3x+b-1x+4x+c       =(−1)1x+2x+a1x+3x+b1x+4x+cOperate R2→R2−R1, R3→R3−R1       =(−1)1x+2x+a01b-a02c-a      =(−1)[c+a−2b]=0                    [∵   a,b,c  are  in  AP⇒2b+a+c]