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(A) Let the first term of the arithmetic progression be $A$ and the common difference be $D$.The terms are given by $a = A + 4D$,$b = A + 7D$,and $c =

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(A) Let the first term of the arithmetic progression be $A$ and the common difference be $D$.The terms are given by $a = A + 4D$,$b = A + 7D$,and $c = A + 12D$.Consider the determinant $\Delta = \begin{vmatrix} a & 5 & 1 \ b & 8 & 1 \ c & 13 & 1 \end{vmatrix}$.Applying row operations $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = \begin{vmatrix} a & 5 & 1 \ b-a & 8-5 & 1-1 \ c-a & 13-5 & 1-1 \end{vmatrix} = \begin{vmatrix} a & 5 & 1 \ 3D & 3 & 0 \ 8D & 8 & 0 \end{vmatrix}$.Expanding along the third column:$\Delta = 1 \cdot [(3D)(8) - (8D)(3)] = 1 \cdot [24D - 24D] = 0$.

If $a, b, c$ are respectively the $5^{\text{th}}, 8^{\text{th}}, 13^{\text{th}}$ terms of an arithmetic progression,then $\left|\begin{array}{ccc}a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1\end{array}\right|=$

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