Proof Involving Legendre Symbol and Quadratic Residue Multiplication

[deroliver]

I am struggling with this proof. If anyone could get me going on it, I would appreciate it!
I recently took a Number Theory class, and we had gotten though Legendre symbols and Quadratic Residues/Non Quadratic Residues.

Anyways, I found this problem in my book, and I was trying to work through it. The result is quite interesting, but I am not sure exactly how it works, or how to begin.

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Let p be an odd prime, prove that:

. . . . .\(\displaystyle \left(\, \dfrac{1\, \cdot\, 2}{p}\, \right)\, +\, \left(\, \dfrac{2\, \cdot\, 3}{p}\, \right)\, +\, \left(\, \dfrac{3\, \cdot\, 4}{p}\, \right)\, +\, ...\, +\, \left(\, \dfrac{(p\, -\, 2)\, (p\, -\, 1)}{p}\, \right)\, =\, -1\)

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If you guys could help me out, that would be awesome!

 

[ksdhart2]

I'd double check that formula. As it stands, that's not going to work, for any positive p, prime or not. You're adding successive fractions, where each one is bigger than the previous, and the smallest one is always positive. So, how can that possibly add up to negative one?