Lesson 25

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.