If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0$, then $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\sum^n_{r=1}\alpha^r+\displaystyle\lim_{n\rightarrow \infty}\displaystyle\sum^n_{r=1}\beta^r$ is equal to?

The sum of the infinite geometric series sum(alpha^r) is alpha / (1 - alpha) and sum(beta^r) is beta / (1 - beta). The sum is (alpha(1-beta) + beta(1-alpha)) / ((1-alpha)(1-beta)) = (alpha + beta - 2*alpha*beta) / (1 - (alpha + beta) + alpha*beta). From 375x^2 - 25x - 2 = 0, alpha + beta = 25/375 = 1/15 and alpha*beta = -2/375. Substituting these: (1/15 - 2(-2/375)) / (1 - 1/15 - 2/375) = (25/375 + 4/375) / (375/375 - 25/375 - 2/375) = (29/375) / (348/375) = 29/348 = 1/12.