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There are a variety of different methods. One of the easiest to use would be having a script that duplicates the records, adding one week to all the dates. However, that requires being able to write the code, or having budget to hire someone to write the code.
Another possibility is to have a formula field that adds one week to the actual date and time. Then right after you duplicate the records, do a bulk copy/paste from the formula field to the actual date and time field. It is kludgy, but it will work.