Lesson 25

Summing Up

Lesson Narrative

In this activity, students use a polynomial identity derived in an earlier lesson, \(x^n-1 = (x-1)(x^{n-1}+ \ldots + x^2+x+1)\), to derive a formula for the sum of the first \(n\) terms in a geometric sequence. While this is commonly known as the formula for the sum of a finite geometric series, student facing language was purposefully written to only refer to sequences since series are not a topic of study in this course.

Students begin by returning to the Koch Snowflake that was first introduced in the previous unit. Thinking of the snowflake as a single triangle with more triangles added at each iteration following a specific pattern, students make sense of this pattern as a series of shapes, as the number of triangles added at each step, and as a general formula for finding the number of triangles added at any given iteration (MP1). Students are then guided to manipulate a general version of the equation for the sum of all the added triangles into a short, rational formula. In the following activity, students shift context to a prescribed drug course, but are still working with a geometric sequence, using the new formula to get a much shorter expression instead of having to add 30 different terms together. In each context, students make connections between the structures of the long form of the sum, \(a(1+r+r^2+ \ldots +r^{n-1})\), and the shorter form of the formula, \(a \frac{1-r^{n}}{1-r}\), using the earlier identity (MP7). In the next lesson, students will continue to practice applying the formula to different situations.

Learning Goals

Teacher Facing

  • Calculate sums of terms in a geometric sequence by using a formula.
  • Understand the derivation of the formula for the sum of the first $n$ terms in a finite geometric sequence.

Student Facing

  • Let’s figure out a better way to add numbers.

Learning Targets

Student Facing

  • I understand why the geometric sum formula is true.

CCSS Standards


Building Towards

Print Formatted Materials

Teachers with a valid work email address can click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.

Student Task Statements pdf docx
Cumulative Practice Problem Set pdf docx
Cool Down Log In
Teacher Guide Log In
Teacher Presentation Materials pdf docx

Additional Resources

Google Slides Log In
PowerPoint Slides Log In