Is there such a thing as backwards sigma? Is there a math function, similar to sigma, that can run down? For example instead of $\sum\limits_{i=1}^{10}i$ , something that adds from 10 to 1 (like a backwards run)...

If $a, b > 0$ then you can say
$$\sum_{k=a}^b a_k = \sum_{k=0}^b a_k - \sum_{k=0}^{a-1} a_k. $$
For example:
\begin{align}
  \sum_{k=3}^7 k &= \sum_{k=0}^7 k - \sum_{k=0}^2 k\\[0.3cm]
    &= (0+1+2+3+4+5+6+7) - (0+1+2)\\[0.3cm]
    &= 3 + 4 + 5 + 6 + 7\\[0.3cm]
    &= 25
\end{align}
And:
\begin{align}
  \sum_{k=7}^3 k &= \sum_{k=0}^3 k - \sum_{k=0}^6 k\\[0.3cm]
    &= (0+1+2+3) - (0+1+2+3+4+5+6)\\[0.3cm]
    &= -(4 + 5 + 6)\\[0.3cm]
    &= -15
\end{align}
