A series of points

The abscissa and ordinates of $n$ points in an arithmetic progression have their first term as $a$ and have common differences of 1 and 2, respectively.

If the algebraic sum of perpendiculars from these points on a variable line which always passes through the point $\left(\dfrac{13}{2}, 11 \right)$ is zero, then evaluate $\left|a-n\right|$ .