"第一期的" can be translated into English as "the first phase (of)" or "the first installment (of)", depending on the context.
If referring to a project, program, or construction phase, "the first phase (of)" would be appropriate. For example, "the first phase of the construction project"（建设项目的第一期）。
If referring to a series of payments, publications, or broadcasts, "the first installment (of)" might be more suitable. For example, "the first installment of the novel series"（小说系列的第一期）。