Proving an identity and some interesting maths stuff

[gentsagree]

So, I would like to prove that

\gamma^{\mu_{1}...\mu_{r}}=(-)^{r(r-1)/2}\gamma^{\mu_{r}...\mu_{1}}

where the matrix gamma is a totally antisymmetric matrix defined as \gamma^{\mu_{1}...\mu_{r}}=\gamma^{[\mu_{1}}\gamma^{\mu_{2}}...\gamma^{\mu_{r}]}

What I have done is to prove that

\gamma^{\mu_{1}...\mu_{r}}=(-)^{(r-1)+(r-2)+...+1}\gamma^{\mu_{r}...\mu_{1}}

by simply commuting all the matrices past each other until their order is reversed (picking up just the minus sign as they are antisymmetrised, so we can take \mu_{i}\neq\mu_{j} for i\neq j).

What's a nice way to see that (r-1)+(r-2)+...+1=r(r-1)/2? It works for some values of r, which one can see by substituting in.

ALSO - PART 2

I am aware of \sum_{n=1}^{\infty}n=\frac{x(x+1)}{2}=-\frac{1}{12},

but I found out that
\int^{1}_{0}\frac{x(x-1)}{2}dx=-\frac{1}{12}

Any comments or clarifications on this relationship between \frac{x(x-1)}{2} and \frac{x(x+1)}{2}.

 

[Avodyne]

You're just summing the positive integers from 1 to $r-1$. Here's how Gauss solved this problem (for $r=101$) when he was in elementary school:
http://mathandmultimedia.com/2010/09/15/sum-first-n-positive-integers/