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<h4>Activity (10 minutes)</h4>
<p>The purpose of this task is to provide students with practice working with geometric sequences and identifying the common ratio of a sequence.</p>
<p>For each sequence, invite a student to share how they completed the sequence and determined the common ratio. Highlight the method of dividing any term by the previous term to find the common ratio. Emphasize that the presence of a common ratio is what makes a sequence a geometric sequence.</p>
<h4>Launch</h4>
<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">Action and Expression: Provide Access for Physical Action</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Provide access to tools and assistive technologies, such as a calculator or graphing software. Some students may benefit from a checklist or list of steps to use the calculator or software. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Conceptual processing; Attention</p>
</div>
</div>
<br>
<h4>Student Activity </h4>
<p>Write the following terms and definitions in your math notebook:</p>
<p>A geometric sequence is a sequence in which each term is found by multiplying the previous term by the same constant.</p>
<p>The number that is multiplied by each term is called the common ratio.</p>
<ol class="os-raise-noindent">
<li>
Complete each geometric sequence.
</li>
<ol class="os-raise-noindent" type="a">
<li>
1.5, 3, 6, _a__, 24, _b__
</li>
<li>
40, 120, 360, __a__, __b__
</li>
<li>
200, 20, 2, __a__, 0.02, __b__
</li>
<li>
\(\frac17\), __a__, \(\frac97\), \(\frac{27}7\), __b__
</li>
<li>
24, 12, 6, __a__, __b__
</li>
</ol>
</ol>
<p class="os-raise-text-bold">Answer:</p>
<ol class="os-raise-noindent" type="a">
<li>
12 , 48 </li>
<li>
1,080, 3,240
</li>
<li>
0.2,
0.002
</li>
<li>
\(\frac37\),
\(\frac{81}7\)
</li>
<li>
3,
1.5
<br>
<br>
</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li>
For each sequence, find its common ratio.
<ol class="os-raise-noindent" type="a">
<li>
1.5, 3, 6, ___, 24, ___
</li>
<li>
40, 120, 360, ___, ___
</li>
<li>
200, 20, 2, ___, 0.02, ___
</li>
<li>
\(\frac17\), ___, \(\frac97\), \(\frac{27}7\), ___
</li>
<li>
24, 12, 6, ___, ___
</li>
</ol>
</li>
<p class="os-raise-text-bold">Answer: </p>
<ol class="os-raise-noindent" type="a">
<li>
2
</li>
<li>
3
</li>
<li>
\(\frac{1}{10}\)
</li>
<li>
3
</li>
<li>
\(\frac12\)
</li>
</ol>
</ol>
<h4>Activity Synthesis</h4>
<p>In the lead up to writing recursive definitions for sequences, it is important for students to understand that for geometric sequences, the common ratio is defined to be the multiplier from one term to the next. Said another way, the common ratio is the quotient of a term and the previous term. For example, many students will want to say that the pattern of the third sequence is “divide by 10 each time.” This is true, but the common ratio is \(\frac1{10}\).</p>
<h4>Video: Completing Geometric Sequences</h4>
<p>Watch the following video to learn more about 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/befe818ae5a7e5b370b57698151b35c39b80849f">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/6b6a6694b54997af858d05e4d65298543c67fe0e" srclang="en_us">https://k12.openstax.org/contents/raise/resources/befe818ae5a7e5b370b57698151b35c39b80849f
</video></div>
</div>
<br>
<h3>4.15.3: 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 geometric?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
5, 10, 15, 20 . . .
</td>
<td>
Incorrect. Let’s try again a different way: The sequence creates terms by adding 5 to the previous term, so there is not a common ratio. The answer is 48, 24, 12, 6, . . .
</td>
</tr>
<tr>
<td>
48, 24, 12, 6, . . .
</td>
<td>
That’s correct! Check yourself: The common ratio is \(\frac12\), so this is a geometric sequence.
</td>
</tr>
<tr>
<td>
3, 5, 8, 10, 13, . . .
</td>
<td>
Incorrect. Let’s try again a different way: The sequence is formed by adding 2, then adding 3, and repeating that pattern. There is not a common ratio. The answer is 48, 24, 12, 6, . . .
</td>
</tr>
<tr>
<td>
10, 7, 4, 1, . . .
</td>
<td>
Incorrect. Let’s try again a different way: The sequence is formed by subtracting 3 from the previous term. There is not a common ratio. The answer is 48, 24, 12, 6, . . .
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.15.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>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.</em></p>
<h4>Finding a Common Ratio</h4>
<h5>Definition of a Geometric Sequence</h5>
<p>A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If \(a_1\) is the initial term of a geometric sequence and \(r\) is the common ratio, the sequence will be</p>
<p>{\(a_1, a_1\cdot r, a_1\cdot r^2, a_1\cdot r^3\), . . .}</p>
<p>How to:</p>
<p>Given a set of numbers, determine if they represent a geometric sequence.</p>
<p>1. Divide each term by the previous term.</p>
<p>2. Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.</p>
<p>Example:</p>
<p>Is the sequence geometric? If so, find the common ratio.</p>
<p>a. 1, 2, 4, 8, 16, . . .</p>
<p>b. 48, 12, 4, 2, . . .</p>
<ol class="os-raise-noindent" type="a">
<li>
Is a geometric sequence. Each term is multiplied by 2, so 2 is the common ratio.
</li>
<li>
This is not geometric because each term is not multiplied by the same constant. There is not a common ratio.
</li>
</ol>
<h4>Try It: Finding a Common Ratio</h4>
<p>Is the sequence geometric? If so, find the common ratio.</p>
<p>100, 20, 4, \(\frac45\), . . .</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 geometric:</p>
<p>Each term is \(\frac15\) of the previous term. So, the common ratio is \(\frac15\), and this is a geometric sequence.</p>