Dedicated to my best friend

Algebra Level 5

For all \(x \in \mathbb C\), let \(S_n=x^n+\frac 1 {x^n}\).

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

This is one of my original Madness problems.
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