Can someone explain this summation to me?

[bry32321]

The sum of x, from x=m to n is equal to -(1/2)(m-n-1)(m+n)

I'm having a hard time figuring out how the sum equals that answer. Can someone explain it to me in detail?

Thank you for any feedback!

 

[daon2]

The standard Guass argument?

\(\displaystyle \begin{matrix} m &+& (m+1) &+& (m+2) &+& \cdots &+& (n-2) &+& (n-1) &+& n\\
n &+& (n-1) &+& (n-2) &+& \cdots &+& (m+2) &+& (m+1) &+& m\end{matrix}
\)

Adding downward makes each equal to \(\displaystyle n+m\). So how many \(\displaystyle (n+m)\)'s are there in total? Note that adding everything up results in two times the requested amount.

 

[bry32321]

Oh okay. Thank you!