Let $t_r$ be the $r^{\mathrm{th}}$ term of an A.P. for $r=1, 2, 3, \dots \dots$ If for some positive integers $m, n$ we have $t_m=\frac{1}{n}$ and $t_n=\frac{1}{m}$, then $t_{mn}$ equals

A particle begins at the origin and moves successively in the following manner as shown, $1$ unit to the right, $1/2$ unit up, $1/4$ unit to the right, $1/8$ unit down, $1/16$ unit to the right etc. The length of each move is half the length of the previous move and movement continues in the 'zigzag' manner indefinitely. The co-ordinates of the point to which the 'zigzag' converges is -