How to Solve a Function Given Some of its Solutions

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Suppose you have a function that defines a series. And suppose you know Some (not all) of the elements of that series. For example, you know your function is n/J, where n is for all positive integers and J is a constant. Your choice for the value of J determines the elements of the resulting series. You also know some of the elements you want to find in that series. How would you find the value of J, given its functions and some of its solutions?