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<h4>Activity (15 minutes)</h4>
<p>The purpose of this activity is for students to contrast three different types of sequences and to introduce the term
“arithmetic sequence.”</p>
<p>Monitor for students using precise language, either orally or in writing, during work time to invite to share during
the whole-class discussion.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.</p>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 2 Collect and Display: Conversing, Speaking, Writing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
As students discuss the sequences, circulate and listen to students talk about the patterns they notice. Write down
common or important phrases you hear students say about each type, for example: “add 10 each time” or
“multiply each term by 2.” Collect the responses into a visual display. Throughout the remainder of the
lesson, continue to update collected student language and remind students to borrow language from the display as
needed. In the lesson synthesis, after the term “arithmetic sequence” is introduced, ask students to
sort the collected language into three groups, one for language used to describe arithmetic sequences, a second for
geometric sequences, and the third for neither.</p>
<p class="os-raise-text-italicize">Design Principle(s): Support sense-making; Maximize meta-awareness</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Representation: Internalize Comprehension</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Activate or supply background knowledge about sequences. Allow students to use calculators to ensure inclusive
participation in the activity. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Memory; Conceptual processing</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Here are the values of the first 5 terms of 3 sequences:</p>
<ol class="os-raise-noindent" type="A">
<li>
30, 40, 50, 60, 70, . . .
</li>
<li>
0, 5, 15, 30, 50, . . .
</li>
<li>
1, 2, 4, 8, 16, . . .
</li>
</ol>
<ol class="os-raise-noindent">
<li>
For each sequence, describe a way to produce a new term from the previous term.
</li>
<p> <strong>Answer: </strong> <br>
A. For example: Each term is 10 more than the previous term.
</p>
<p>
B. For example: Each term is the result of adding 10 more than was added to the previous term (add 5, then add 10,
and then add 15).
</p>
<p>
C. For example: Each term is double the previous term.
</p>
<ol class="os-raise-noindent" type="A">
<li>
30, 40, 50, 60, 70, . . .
</li>
<li>
0, 5, 15, 30, 50, . . .
</li>
<li>
1, 2, 4, 8, 16, . . .
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="2">
<li>
If the patterns you described continue, which sequence has the second-greatest value for the 10th term?
</li>
<p> <strong>Answer: </strong>
Sequence<span> B</span></p>
</ol>
<ol class="os-raise-noindent" start="3">
<li>
Which of these could be geometric sequences? Explain how you know.
</li>
<p> <strong>Answer: </strong>
Sequence <span>C</span>. For example: A geometric sequence has the same multiplier from one term to the next, the
common ratio, and neither <span>A </span>nor<span> B</span> has this. Sequence <span>C </span>does; the common ratio
is 2. </p>
</ol>
<p>An arithmetic sequence is a sequence in which each term is the previous term plus a constant.</p>
<p>This constant is called the common difference.</p>
<ol class="os-raise-noindent" start="4">
<li>Which of these could be arithmetic sequences? Explain how you know.</li>
<p> <strong>Answer: </strong><span>Sequence <span>A</span>. For example: An arithmetic sequence gets each term by
adding the same constant to the previous term. Sequence<span> A</span> adds 10 to each term to find the next
term.</span></p>
</ol>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>Elena says that it’s not possible to have a sequence of numbers that is both arithmetic and geometric. Do you
agree with Elena? Be prepared to show your reasoning.</p>
<p class="os-raise-text-bold">Answer: </p>
<p>Elena is incorrect. Sample explanation: the sequence 1, 1, 1, . . . is arithmetic (adding 0 each time), but it is
also geometric (multiplying by 1 each time).</p>
<h4> Video: Comparing Sequences</h4>
<p>Watch the following video to learn more about comparing arithmetic and geometric sequences.
</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/509740b91324e860a23860c580fa239d8a51c4dd">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/12c5ed810194f2f3cdae39856f3db5a28cd0d3c8" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/509740b91324e860a23860c580fa239d8a51c4dd
</video></div>
</div>
<br>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Students may need help identifying a pattern for the two non-geometric sequences. Ask, “What can you say about
the change between consecutive terms in the sequence?” Encourage students to use any language they wish to
describe the pattern—they do not need to use an equation at this time. Some students may benefit from creating a
table where they can see the term number and the value of the term side by side, particularly for Sequence B.</p>
<h4>Activity Synthesis</h4>
<p>The purpose of this discussion is to compare different types of sequences and introduce students to the term
“arithmetic sequence.” Begin the discussion by asking students how <span>A and C are alike and
different. They might offer things like:</span></p>
<ul>
<li>
C is geometric, but A is not.
</li>
<li>
In A, you always add 10 to get from term to term, but in C, you always multiply by 2.
</li>
<li>
In C, the growth factor is 2. In A, you get the next term by adding 10 to the previous term.
</li>
</ul>
<p>Tell students that sequence A is an example of an arithmetic sequence. Here are two ways to know a sequence is
arithmetic:</p>
<ul>
<li>
You always add the same number to get from one term to the next.
</li>
<li>
If you subtract any term from the next term, you always get the same number.
</li>
</ul>
<p>Share that the constant in an arithmetic sequence is called the rate of change or common difference. In sequence A,
the rate of change is 10 because \(40=30+10\), \(50=40+10\), \(60=50+10\), and \(70=60+10\).</p>
<p>Some students may notice the similarity between an arithmetic sequence and a linear function. Invite these students
to share their observations, such as how both are defined by a constant rate of change. Tell students that arithmetic
sequences are a type of linear function and that their knowledge of linear functions will help them describe
arithmetic sequences during this unit. If students do not bring up the connection to linear functions, ask:
“What do you remember about linear functions?” Record student responses for all to see and invite
comparisons between linear functions and arithmetic sequences.</p>
<h3>4.16.2: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts
explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Which of the following sequences is arithmetic?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
3, 4, 6, 9, 13, . . .
</td>
<td>
Incorrect. Let’s try again a different way: While a constant is added to the previous term to generate
the next term, the constant is different each time. The answer is 3, 7, 11, 15, . . .
</td>
</tr>
<tr>
<td>
1, 4, 16, 64, …
</td>
<td>
Incorrect. Let’s try again a different way: This sequence is geometric with a common ratio of 4. The
answer is 3, 7, 11, 15, . . .
</td>
</tr>
<tr>
<td>
3, 7, 11, 15, . . .
</td>
<td>
That’s correct! Check yourself: To find the next term, 4 is added to the previous term. This is
arithmetic.
</td>
</tr>
<tr>
<td>
25, 5, 1, \(\frac15\), . . .
</td>
<td>
Incorrect. Let’s try again a different way: This sequence has a common ratio of \(\frac15\) and is
geometric. The answer is 3, 7, 11, 15, . . .
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.16.2: Additional Resources</h3>
<p class="os-raise-text-bold">The following content is available to students who would like more support based on their experience with
the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it.</p>
<h4>Arithmetic Sequences</h4>
<p class="os-raise-text-bold">Finding Common Differences</p>
<p>Companies often make large purchases, such as computers and vehicles, for business use.</p>
<ul>
<li>
A woman who starts a small contracting business purchases a new truck for $25,000.
</li>
<li>
After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck
will therefore be $17,000, which is $3,400 per year for five years.
</li>
<li>
The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400
after four years; and $8,000 at the end of five years.
</li>
</ul>
<p>The values of the truck in the example are said to form an arithmetic sequence because they change by a constant
amount each year. Each term increases or decreases by the same constant value called the common difference of the
sequence. For this sequence, the common difference is –3,400.</p>
<p><img alt="ARITHMETIC SEQUENCE WITH SIX TERMS WITH A COMMON DIFFERENCE OF NEGATIVE THREE THOUSAND FOUR HUNDRED, STARTING WITH A VALUE OF TWENTY-FIVE THOUSAND AND ENDING WITH A VALUE OF EIGHT THOUSAND.
" src="https://k12.openstax.org/contents/raise/resources/b4ef05a451ee9ddbd41a3de6a0018926dcdbd443" width="550"></p>
<p>The sequence below is another example of an arithmetic sequence. In this case, the common difference is 3. You can
choose any term of the sequence and add 3 to find the subsequent term.</p>
<p><img alt="ARITHMETIC SEQUENCE WITH A COMMON DIFFERENCE OF THREE, WRITTEN AS THREE, SIX, NINE, TWELVE, FIFTEEN, DOT DOT DOT.
" src="https://k12.openstax.org/contents/raise/resources/ad2abbd7eab8709bee9fc741c600c38ac25e82a6"></p>
<p class="os-raise-text-bold">ARITHMETIC SEQUENCE</p>
<p>An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a
constant. This constant is called the common difference. If \(a_1\) is the first term of an arithmetic sequence and
\(d\) is the common difference, the sequence will be:</p>
<p>{\(a_n\)}={\(a_1, a_1+d, a_1+2d, a_1+3d\), . . .}</p>
<p>Example:</p>
<p>Is each sequence arithmetic? If so, find the common difference.</p>
<ol class="os-raise-noindent" type="a">
<li>
{1, 2, 4, 8, 16, . . .}
</li>
<li>
{-3, 1, 5, 9, 13, . . .}
</li>
</ol>
<p>Subtract each term from the subsequent term to determine whether a common difference exists.</p>
<ol class="os-raise-noindent" type="a">
<li>
The sequence is not arithmetic because there is no common difference.
<br>
<br>\(2-1=1\) \(4-2=2\) \(8-4=4\) \(16-8=8\)
</li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
<li>
The sequence is arithmetic because there is a common difference. The common difference is 4.
<br>
<br>\(1-(-3)=4\) \(5-1=4\) \(9-5=4\) \(13-9=4\)
</li>
</ol>
<br>
<h4>Try It: Arithmetic Sequences</h4>
<p>Is the given sequence arithmetic? If so, find the common difference.</p>
<p>{18, 16, 14, 12, 10, . . .}</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to determine if the sequence is arithmetic:</p>
<p>\(16-18=-2\)<br>
\(14-16=-2\)<br>
\(12-14=-2\)<br>
\(10-12=-2\)</p>
<p>This is an arithmetic sequence with a common difference of –2.</p>