Dedicated to my best friend

Algebra Level 5

For all xCx \in \mathbb C, let Sn=xn+1xnS_n=x^n+\frac 1 {x^n}.

After removing a number SbS_b from the series (S1+S2++Sa)(S_1+S_2+\ldots+S_a) when Sandeep Bhardwaj, the magician, divides the remaining part with Sb,S_b, he somehow always manages to get another series (S1+S2++Sc)(S_1+S_2+\ldots+S_c) as the answer. Now find the value of (abc)(a-b-c) and therefore, decipher his trick.

This is one of my original Madness problems.
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