This Warm-up asks students to calculate the sum of the first 50 terms of two different sequences, but with less scaffolded support than in previous work. The goal of this activity is to build student flexibility in using the formula for the sum and thinking about the terms in a sequence by comparing two related sequences, specifically, two sequences in which one has values double that of the first.

Monitor for students who identify the sum of the second sequence in different ways, such as by:

• Thinking of it as a sequence with the same - and -values as the first but a different -value
• Noticing it must be twice the sum of the first
• Seeing it as 1 plus the sum of the first sequence, where the first sequence has instead of

The goal of this discussion is that students see the relationship between both the two sequences and in how the sum of the first 50 terms of each sequence are calculated.

Once students are agreed on the -, -, and -values for the first sequence, display 2–3 approaches for the second sequence from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What kinds of additional details or language helped you understand the approaches? Are there any additional details or language that you have questions about?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Why do the different approaches lead to the same outcome?”

If not brought up by a student for the second strategy, display the relationship as one way to understand why the second sequence has a sum twice as large.

In this activity students are asked to use the formula for the sum of the first terms in a geometric series in a context involving percent change. Of particular focus in the activity is students identifying the correct -value when applying the formula and interpreting the meaning of their calculations in the given context of home sales.

Monitor for students who can clearly articulate that the formula in the second question is for the number of houses sold from 2010 to 2020, not from 2010 to 2021, because 2010 is included in the years being summed.

This activity returns students to a context they first looked at at the start of the “Polynomial Functions” unit: money invested yearly into a savings account with a fixed interest rate that compounds annually. While the situation described in the task is a simplified version of how a modern savings account actually works, which is something students will investigate further in a future unit, the goal of this task is for students to use the formula to better understand the nature of situations that can be described with geometric series. In particular, students should see the power of a common ratio greater than 1 to increase what may seem like relatively small values over time.

Before students begin working, they are explicitly asked to make an estimate. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).

Monitor for any students who choose to use graphing technology to answer the last question. Making graphing technology available gives students opportunity to choose appropriate tools strategically (MP5).