Let $\Delta=\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}
x+2 & x+3 & x+2 a \\
0 & 0 & 2(2 b-a-c) \\
x+4 & x+5 & x+2 c
\end{array}\right|$
(using $\mathrm{R}_{2} \rightarrow 2 \mathrm{R}_{2}-\mathrm{R}_{1}-\mathrm{R}_{3}$ )
But $\mathrm{a}$, b and c are in APusing $2 \mathrm{~b}=\mathrm{a}+\mathrm{c}$, we get
$\Delta=\frac{1}{2}\left|\begin{array}{ccc}
\mathrm{x}+2 & \mathrm{x}+3 & \mathrm{x}+2 \mathrm{a} \\
0 & 0 & 0 \\
\mathrm{x}+4 & \mathrm{x}+5 & \mathrm{x}+2 \mathrm{c}
\end{array}\right|=0$
Since, all elements of $\mathrm{R}_{2}$ are zero.