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We formulate and analyze a generic sequential resource access problem arising in a variety of engineering fields, where a user disposes a number of heterogeneous computing, communication, or storage resources, each characterized by the probability of successfully executing the user's task and the related access delay and cost, and seeks an optimal access strategy to maximize her utility within a given time horizon, defined as the expected reward minus the access cost. We develop an algorithmic framework on the (near-)optimal sequential resource access strategy. We first prove that the problem of finding an optimal strategy is NP-hard in general. Given the hardness result, we present a greedy strategy implementable in linear time, and establish the closed-form sufficient condition for its optimality. We then develop a series of polynomial-time approximation algorithms achieving $(ε,δ)$-optimality, with the key component being a pruning process eliminating dominated strategies and, thus maintaining polynomial time and space overhead.