The purpose of this lesson is for students to understand what makes a sequence geometric and to begin to connect that idea with their learning about exponential functions in an earlier course.

A geometric sequence is a function in which the terms of the sequence grow by the same factor from one term to the next. For example, in the geometric sequence 0.5, 2, 8, 32, 128, . . . , each term is 4 times the previous term. Two ways to think about how you know the sequence is geometric are:

• Each term is multiplied by a factor of 4 to get the next term.
• The ratio of each term and the previous term is 4.

We call 4 the growth factor or the common ratio.

Students begin the lesson articulating what they notice and wonder (MP6) about examples of geometric sequences. Next, they develop two different sequences from the context of repeatedly cutting a piece of paper in half. Students use tables and graphs to identify the growth factor and define these geometric sequences. Lastly, students practice calculating missing terms of geometric sequences, using repeated reasoning (MP8) to make sense of the sequence and calculate the growth factor.

In an earlier course, students encountered the idea of functions and studied exponential functions specifically. Some students may see that a geometric sequence is simply an exponential function whose outputs are the terms and whose inputs are the positions of the terms. The Lesson Synthesis invites students to recall those ideas with a light touch by referring to, for example, “the size of each piece as a function of the number of cuts” and by using the term “growth factor” rather than “common ratio.” Students will have more opportunities in future lessons to make connections to exponential functions.