Before I grind some algebra I was wondering if there was a known equation for a series of the form:
$$(x-y)+(x-2y)+(x-3y)+\dots+(x-ny) = T$$
Also a variant:
$$T-(q+y)-(q+2y)-(q+3y)-\dots-(q+ny) = 0$$
The first goes from $0$ to $T$ and the second from $T$ to $0$.
$x$, $y$, and $T$ are known for each instance, but change with each instance.