1000 Followers problem!

Algebra Level 5

\[\large \alpha^{\eta} - \alpha^2-\alpha-\eta=0\]

Consider the above polynomial and let its roots be \(\large\beta_{(1,\eta)},\beta_{(2,\eta)},\beta_{(3,\eta)} , \ldots, \beta_{(\eta,\eta)}\).

Define \(\large S_{\eta}= \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^{\eta}\), find the value of \(\large \displaystyle\sum_{j=5}^{1000} S_j\).

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