Lesson 5
Sequences are Functions
Lesson Narrative
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. In previous lessons, they described the arithmetic sequence 99, 96, 93, . . . as starting at 99 where each term is 3 less than the previous term. Now they think of it as a function \(f\) of the position, starting at position 1, and write \(f(1)=99\) and \(f(n)=f(n1)3\) for \(n\ge2\) , where \(n\) is an integer. 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. Students will use recursive definitions to describe functions in both mathematical and realworld contexts throughout the remainder of this unit. It is not necessary that they use the term "recursive definition" however.
In the warmup, students make sense of a dot pattern as a function where the number of dots in each step depends on the step number (MP1). 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). 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. Students then return to sequences they have seen previously in the unit and define them recursively using function notation.
Regarding the notation students use when writing an equation that defines a sequence, avoid being overly prescriptive. It is more important that students can encapsulate the rule correctly than it is that they can do it in a particular format. Throughout this unit, definitions of sequences written in function notation are always followed by an inequality using \(n\) that describes the domain of the function, which is particularly helpful when students work with definitions for the \(n^{\text{th}}\) term of a sequence and need to indicate what the first term of the sequence is. This is not meant to imply that all students should use such inequalities every time they define a sequence using function notation. In your classroom, it may be better to
 Ask students to write out in words, instead of inequalities, the restrictions on the domain of a sequence.
 Keep the focus on writing equations and tell students to be prepared to respond orally when asked about the domain of a sequence.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems, such as by using a spreadsheet to generate sequences. We recommend making technology available (MP5).
Learning Goals
Teacher Facing
 Comprehend that sequences are functions whose domain is a subset of the integers.
 Create (in writing) a recursive definition for a sequence using function notation.
Student Facing
 Let's learn how to define a sequence recursively.
Learning Targets
Student Facing
 I can define arithmetic and geometric sequences recursively using function notation.
CCSS Standards
Glossary Entries

arithmetic sequence
A sequence in which each term is the previous term plus a constant.

geometric sequence
A sequence in which each term is a constant times the previous term.