Don't mind Plato, he gets cranky sometimes.

Your formula for $\displaystyle S_n$ for the tortoise looks very strange. I think you have an extra divisor $\displaystyle 2$ in there.

$\displaystyle S_n= \frac12{n\big(2a+(n-1)\big)}$

looks better to me. It also gives the correct formula that you suggested. Your formula for the hare is also correct.

Your series are OK too. Plato just used the more advanced sigma notation (but his formula for the hare is wrong (as far as I can tell). The sigma notation involves writing a formula the generates each term of the series separately, and then using the $\displaystyle \sum$ sign and its subscript and superscript to show which particular elements of the series we are summing.

Your formulas just show the distance travelled after $\displaystyle n$ minutes (as required). You are correct that the way in which the question is framed means that the hare never gets to the end of the race. It's an example of Zeno's paradox, except that Zeno used Achilles and a tortoise for his thought experiment.