Is there a large counterexample?

Algebra Level 4

True or False?

If \(a,b\) and \(c\) are roots to the equation \(x^3+x^2 + x=1\), and let \(S_n = a^n + b^n + c^n\), then there exists a positive integer \(n\) such that \(\text{sgn}(S_n) = \text{sgn}(S_{n+1}) = \text{sgn}(S_{n+2}) \).

\[\] Notation: \(\text{sgn}(x) := \begin{cases} \begin{array} {l l } -1 & \text{ if }x<0 \\ 0 & \text{ if }x=0 \\ 1 & \text{ if }x>0 \\ \end{array} \end{cases} \) denotes the sign function.

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