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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.

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    These are sums of arithmetic progressions, so yes, there are standard formulas.2012-05-10
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    $T$ is not the same in those two expressions, and if they are then $q=x-(n+1)y$2012-05-13

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