Not Limiting a Sum

Algebra Level 4

\(\alpha, \beta\) are the roots of \(375x^2-25x-2=0 \). Denote \( S_n = \alpha^n+\beta^n \), and if the summation below is in the form of \( \frac a b \) where \(a,b\) are coprime positive integers. Find the value of \(a+b\).

\[\large \displaystyle \sum_{n=1}^\infty S_n \]

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