1000 Followers problem!

Algebra Level 5

αηα2αη=0\large \alpha^{\eta} - \alpha^2-\alpha-\eta=0

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

Define Sη=i=1ηβ(i,η)η\large S_{\eta}= \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^{\eta}, find the value of j=51000Sj\large \displaystyle\sum_{j=5}^{1000} S_j.

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