Is there a large counterexample?

Algebra Level 4

True or False?

If a,ba,b and cc are roots to the equation x3+x2+x=1x^3+x^2 + x=1, and let Sn=an+bn+cnS_n = a^n + b^n + c^n, then there exists a positive integer nn such that sgn(Sn)=sgn(Sn+1)=sgn(Sn+2)\text{sgn}(S_n) = \text{sgn}(S_{n+1}) = \text{sgn}(S_{n+2}) .

Notation: sgn(x):={1 if x<00 if x=01 if x>0\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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