Students then return to sequences they have seen previously in the unit and define them recursively using function notation. Also during this activity, students decide what values make sense for the domain of the function, which leads to expanding their definition of sequence to a function whose domain is a subset of the integers. This helps prepare students to write a recursive definition for the function by expressing regularity in repeated reasoning while using a table in the following activity (MP8). In the warm-up, students make sense of a dot pattern as a function where the number of dots in each step depends on the step number (MP1). It is not necessary that they use the term "recursive definition" however.
Students will use recursive definitions to describe functions in both mathematical and real-world contexts throughout the remainder of this unit. This is called a recursive definition for \(f\) because it describes a repeated, or recurring, process for getting the values of \(f\), namely the process of subtracting 3 each time. Now they think of it as a function \(f\) of the position, starting at position 1, and write \(f(1)=99\) and \(f(n)=f(n-1)-3\) for \(n\ge2\), where \(n\) is an integer. as starting at 99 where each term is 3 less than the previous term. In previous lessons, they described the arithmetic sequence 99, 96, 93. Building on the informal language students have used so far in the unit, the purpose of this lesson is for students to understand that sequences are functions and to use function notation when defining them with equations.
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