Yay for 2014! #6

Algebra Level 5

Let \(r_{1}, r_{2}, ..., r_{2014}\) be the complex roots, not all necessarily distinct, of the polynomial \(x^{2014}+2014x^{2013}-1\). If

\[S = \sum_{n=1}^{2014}\sum_{k=1}^{2014}r_{k}^{n},\]

and \(\frac{2015}{2014}S = a^{2014} + b\), for some positive integers \(a\) and \(b\) such that \(a\) is as large as possible, find the units digit of \(a + b + 2014.\)

This problem is part of the set Yay for 2014!.


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