Not Limiting a Sum

Algebra Level 5

$\alpha, \beta$ 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 $a+b$ .

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