In this section, students learn about sequences. Focusing on geometric and arithmetic sequences, students determine missing values, calculate growth factors and rates of change, and represent sequences using tables and graphs.

The last lesson of this section is optional. It provides activities to teach students how to use spreadsheets and graphing technology, which are useful for working with sequences and give students more options for choosing appropriate tools strategically in their future work.

A note on language and notation: In earlier courses, students may have learned that a ratio is an association between two or more quantities. In more advanced work, such as this course, “ratio” is typically used as a synonym for “quotient.” This expanded use of the word “ratio” comes into play in this section with the introduction of the term “common ratio.”

In this section, students learn that sequences are functions and how to use function notation to define sequences recursively. To strengthen their understanding of recursive definitions, students practice representing sequences in familiar ways with tables and graphs. Throughout this section, students exchange questions, ideas, and strategies with their classmates.

1	20.25
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Graph of a discrete function, H, origin “O”. Horizontal axis, "n," with scale from 0 to 6, by 1’s. Vertical axis, with scale 0 to 24 by 1’s. Coordinates plotted (1 comma 20 point 25), (2 comma 13 point 5), (3 comma 9), (4 comma 6), (5 comma 4).  

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Regarding the notation students use when writing an equation that defines a sequence, avoid being overly prescriptive. It is more important that students can describe 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 that describes the domain of the function. This is particularly helpful when students work with definitions for the 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 inequalities every time they use function notation to define a sequence. 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.

In the final section, students study the domain of a function, explore types of equations, and write equations to define sequences given tables, visual patterns, and situations.

Students begin by moving from recursive to explicit definitions of sequences and continue making connections between sequences and functions. Students also explore how changing the domain can change the equation for the sequence, and they consider reasonable domains for given contexts.

Students also consider how the choice between recursive and explicit equations depends on what they are trying to do and that some representations are more efficient than others for achieving their goals.

Next, students practice writing equations for situations, both real-world and abstract, that can be represented by sequences, and they use the models they create to answer questions.

A three step pattern, black squares surrounded by white squares. “Step 1”, 2 black squares in a row, surrounded by 10 white squares. “Step 2”, a 2 by 3 rectangle made of 6 black squares, surrounded by 14 white squares. “Step 3”, a 3 by 4 rectangle made of 12 black squares, surrounded by 18 white squares.