Lesson 26

Using the Sum

26.1: Some Interesting Sums (5 minutes)

Warm-up

This warm-up picks up where the previous lesson ended asking students to calculate the sum of the first 50 terms of two different sequences, but with less scaffolded support. 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 where 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 $$r$$ and $$n$$ values as the first but a different $$a$$ value.
• noticing it must be twice the sum of the first.
• seeing it as 1 plus the sum of the first sequence, only where the first sequence has $$n=49$$ instead of $$n=50$$.

Student Facing

Recall that for any geometric sequence starting at $$a$$ with a common ratio $$r$$, the sum $$s$$ of the first $$n$$ terms is given by $$s=a\frac{1-r^{n}}{1-r}$$. Find the approximate sum of the first 50 terms of each sequence:

1. $$\frac12$$, $$\frac14$$, $$\frac18$$, $$\frac{1}{16}$$, . . .
2. 1, $$\frac12$$, $$\frac14$$, $$\frac18$$, $$\frac{1}{16}$$, . . .

Student Response

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Activity Synthesis

Once students are agreed on the $$a$$, $$r$$, and $$n$$ values for the first sequence, invite previously selected students to share their work for the second sequence. While students describe their thinking, record any expressions used for all to see. If not brought up by a student for the second strategy, display the relationship $$1 \frac{1-\left(\frac12\right)^{n}}{1-\frac12} = 2 \boldcdot \frac12 \frac{1-\left(\frac12\right)^{n}}{1-\frac12}$$ as one way to understand why the second sequence has a sum twice as large.

26.2: That’s a lot of Houses (15 minutes)

Activity

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

Listen for students who can clearly articulate why the formula in the second question is for the number of houses sold from 2012 to 2022 and not 2012 to 2023, since 2012 + 11 = 2023, to share during the discussion.

Student Facing

In 2012, about 71 thousand homes were sold in the United Kingdom. For the next 3 years, the number of homes sold increased by about 18% annually. Assuming the sales trend continues,

1. How many homes were sold in 2013? In 2014?
2. What information does the value of the expression $$71 \frac{(1-1.18^{11})}{(1-1.18)}$$ tell us?
3. Predict the total number of house sales from 2012 to 2017. Explain your reasoning.
4. Do these predictions seem reasonable? Explain your reasoning.

Student Response

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Student Facing

Han and Lin each have a method to calculate $$3^5 + 3^6 + \ldots + 3^n$$. Han says this is $$3^5 \left(1+ 3+ 3^2 + \ldots +3^{n-5}\right)$$ and concludes that $$3^5+\ldots + 3^n = 3^5\frac{3^{n-4} -1}{3-1}$$. Lin says that this is a difference of terms in 2 geometric sequences and can be written as  $$\frac{3^{n+1}-1}{3-1} - \frac{3^5-1}{3-1}$$. Do you agree with either Han or Lin? Explain your reasoning.

Student Response

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Activity Synthesis

Select students previously identified to share how many homes they determined were sold in 2013 and 2014, and how they reasoned about what the value of $$71 \frac{(1-1.18^{11})}{(1-1.18)}$$ means, specifically attending to why $$n=11$$ means 2012 to 2022 and not 2012 to 2023, which is a common misconception.

Conclude the discussion by informally polling the class to see if they think the prediction for the total number of house sales from 2012 to 2017 seems reasonable. Select students from each side to explain their reasoning. Since these dates are in the past, we can look up home sales data in the United Kingdom and see that after 2015, the data does not continue the trend of increasing by about 18% each year. While there is a spike in sales in 2016 due to a change of law, the sale per year leveled out for several years after that.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . . .” or “I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making

26.3: Back to Funding the Future (15 minutes)

Activity

This activity returns students to a context they first looked at at the start of this 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, the power of a common ratio $$r$$ 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).

Launch

Display the situation in the activity for all to see and ask if students have any clarifying questions. Make sure students understand that the money is deposited at the start of the year, and at the end of the year the interest is added into the account.

Ask students to estimate: “Before calculating anything, predict how much you need to put into the account at the start of each year to have over $100,000 in it when you turn 70”. Poll the class for their estimates and record these for all to see throughout the activity. Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “I think I should use . . . to find out . . . .”, “I noticed . . . so . . . .”, and “Why did you . . . ?” Supports accessibility for: Language; Social-emotional skills Student Facing Let’s say you open a savings account with an interest rate of 5% per year compounded annually and that you plan on contributing the same amount to it at the start of every year. 1. Predict how much you need to put into the account at the start of each year to have over$100,000 in it when you turn 70.
2. Calculate how much the account would have after the deposit at the start of the 50th year if the amount invested each year were:
1. $100 2.$500
3. $1,000 4.$2,000
3. Say you decide to invest $1,000 into the account at the start of each year at the same interest rate. How many years until the account reaches$100,000? How does the amount you invest into the account compare to the amount of interest earned by the account?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

If students are not sure how to begin the second problem, encourage them to write out the value in the account after 5 years to help see the pattern.

Activity Synthesis

Select students to share how they calculated the values of the accounts in the question about what happens when different amounts of money are invested each year, focusing in particular on how students identified the $$a$$, $$r$$, and $$n$$ values before using the formula $$s = a \frac{1-r^{n}}{1-r}$$.

Select previously identified students to share how they used a graph of $$s = 1000 \frac{1-1.05^{n}}{1-1.05}$$ to figure out how many years $$n$$ until the account reaches $100,000. If students did not use a graph, display one for all to see and invite students to identify how to use the graph to answer the last question. Conclude the discussion by returning to the poll results and invite students to calculate how much they would need to invest each year in this situation (about$478 per year). Invite students to share why they think they under- or over-estimated the value they would need to invest each year. The goal of this final discussion is for students to have opportunity to articulate the ideas they have about how geometric sequences change and to make connections between context, formula, and graph where possible.

Lesson Synthesis

Lesson Synthesis

The goal of this synthesis is for students to summarize in their own words what they have learned about using the formula $$s = a \frac{1-r^{n}}{1-r}$$. Display the following writing prompt for all to see: “If someone asked you to explain how to find the sum of the first $$n$$ terms of a geometric sequence, how would you explain it to them?” Encourage students to use examples from the activities or to come up with their own example. If time allows, invite 2–3 students to share their explanations.

26.4: Cool-down - That’s a Lot! (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Let’s say you plan to invest \$200 at the start of each year into an account that averages 3% interest compounded annually at the end of the year. How many years until the account has more than \$10,000? \$20,000? We know that at the end of year 1 the amount in the account is \$206. At the end of year 2 the amount in the account is \$418.18 since $$200(1.03)^2 + 200(1.03)=418.18$$. At the start of year 30, for example, that original \$200 has been compounded a total of 29 times while the last \$200 deposited has been compounded 0 times. Figuring out how much is in the account 30 years after the first deposit means adding up $$200(1.03)^{29} + 200(1.03)^{28} + . . . + 200(1.03) + 200$$. We can use the formula for the sum of a geometric sequence, $$s=a\frac{(1-r^{n})}{(1-r)}$$, to find the total amount in the account. The sequence starts at $$a=200$$ and increases at a rate of $$r=1.03$$ each year. After $$n$$ years, the total $$s$$ in the account is $$s=206\frac{(1-1.03^{n})}{(1-1.03)}$$. Now we have a simpler expression to evaluate for different $$n$$ values. It turns out that when $$n=31$$, the account has about \$10,301 in it and when $$n=47$$, it has about \\$20,682 in it.