In this activity, students use a polynomial identity derived in an earlier lesson, \(x^n1 = (x1)(x^{n1}+ \ldots + x^2+x+1)\), 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, \(a(1+r+r^2+ \ldots +r^{n1})\), and the shorter form of the formula, \(a \frac{1r^{n}}{1r}\), using the earlier identity (MP7). In the next lesson, students will continue to practice applying the formula to different situations.
Lesson overview
 25.1 Warmup: Notice & Wonder: A Snowflake’s Return (5 minutes)

25.2 Activity: A Geometric Addition Formula (20 minutes)
 Includes "Are you Ready for More?" extension problem
 25.3 Activity: The Sum of Antibiotics (10 minutes)
 Lesson Synthesis
 25.4 Cooldown: After the Last Dose (5 minutes)
Learning goals:
 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.
Learning goals (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.
Standards:
 This lesson builds towards the standard: CCSS.HSASSE.B.4MS.A.SSE.4
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