I graded a test for my Calculus II class. One of the topics was infinite sequences. I gave a one point extra credit problem. I wanted them to understand what a recursive definition was and how it differed from a closed form (formula) definition. The problem said, “Give an example of a sequence that is defined recursively, but also has a closed form representation. Give both representations.”

The student gave me the closed form: {n}. The recursive definition gives the first value as one and defines the n + 1st value as 1 added to the nth value. It is a unusually good example because it shows the distinction without undue complications. I had not thought of such a simple yet elegant example. Sometimes students surprise me and that is a good thing.

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Tags: closed form, Education, math, recursive

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