In this paper, we present fast and accurate approximations for the probability of ruin over a finite number of periods, assuming inhomogeneous independent claim size distributions and arbitrary premium income in subsequent periods. We develop exact recursive expressions for the non-ruin probabilities in subsequent periods. These recursive expressions provide the basis for a computationally efficient recursive approximation scheme based on two-moment gamma distribution fits. An extensive simulation experiment showed the accuracy of the approximations for values of the horizon up to 20. Having shown the validity of the approximations, we applied them to gain insight into two non-stationary investment problems.