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<h4>Activity</h4>
<p>The <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">recursive definition</span> is a function that provides a repeated, or recurring, process to find terms in a sequence.</p>
<p>Start with the recursive definition: \(R(n)=3R(n-1) + 1\), \(n\geq2\) and given \(R(1)=-1\). </p>
<p><img src="https://k12.openstax.org/contents/raise/resources/785426b7ff9f6cb449e2011a037825d4ec55187f" width="400"></p>
<ul>
<li> The first part of the recursive definition provides the instructions for how to generate the sequence. It provides the function rule or formula for the nth term. </li>
<li>The second part states the domain of the sequence. It describes where we start and the values that are substituted into the function.</li>
<li>The third part is the basis for the sequence or the smallest term value. </li>
</ul>
<p>See the table below for how to substitute to find more terms in the sequence using the recursive definition. Substitute the term number, \(n\), in for \(n\) in the recursive definition. Complete the table with a partner.</p>
<table class="os-raise-wideadjustedtable">
<thead>
<tr>
<th scope="col">Term number (n)</th>
<th scope="col">Substitute in \(R(n)\)</th>
<th scope="col">Value</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>given</td>
<td>-1</td>
</tr>
<tr>
<td>2</td>
<td>\(3R(2-1) + 1 = 3R(1) + 1 = 3(-1) + 1 = -2\)</td>
<td>-2</td>
</tr>
<tr>
<td>3</td>
<td>\(3R(3-1) + 1 = 3R(2) + 1 = 3(-2) + 1 = -5\)</td>
<td>-5</td>
</tr>
<tr>
<td>4</td>
<td>\(3R(4-1) + 1 = 3R(3) + 1 = 3(-5) + 1\)</td>
<td>a. ___</td>
</tr>
<tr>
<td>5</td>
<td>a. ___</td>
<td>b. ___</td>
</tr>
<tr>
<td>6</td>
<td>b. ___</td>
<td>c. ___</td>
</tr>
</tbody>
</table>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="d51f56fc-2cdc-4f99-84de-1186c0c98346" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Using the recursive definition:
<ol class="os-raise-noindent" type="a">
<li>Substitute \(n=5\) in \(R(n)\)</li>
</ol>
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(3R(5-1)+1\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ec844c23-b7a6-4ff9-9002-f874b7398916" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>Substitute \(n=6\) in \(R(n)\)</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(3R(6-1)+1\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="efc5efd2-36df-4ac7-b309-b08ef39a8841" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Determine the value for each term:<ol class="os-raise-noindent" type="a">
<li>Value when \(n=4\)</li>
</ol>
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>–14</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1d4cde96-7dd5-4439-81db-761a55f96ac7" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>Value when \(n=5\)</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>–41</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="642d1f87-4e4d-4da1-aae9-485862d65e5d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li>Value when \(n=6\)</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>–122</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ef22a3c9-dc7c-4c6e-b53c-ffeec5554d08" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>Write the full sequence out to 6 terms.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>–1, –2, –5, –14, –41, –122</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="232aad94-fb39-45d7-8f2d-05b1b3e1f3cf" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1-->
<div class="os-raise-ib-pset-problem" data-content-id="8482cb5d-ffd9-4df8-a1ea-03a368002bbc" data-problem-type="dropdown" data-solution="Geometric" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="4">
<li>Determine if the following sequences are arithmetic or geometric.
<ol class="os-raise-noindent" type="a">
<li>3, 6, 12, 24</li>
</ol>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Geometric </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Geometric. </p>
</div>
</div>
<!--END QUESTION.-->
<br>
<!--Q2-->
<div class="os-raise-ib-pset-problem" data-content-id="e4499a8e-3c8c-4333-ac8d-a11528814f06" data-problem-type="dropdown" data-solution="Geometric" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2">
<ol class="os-raise-noindent" start="2" type="a">
<li>18, 36, 72, 144</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Geometric </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Geometric. </p>
</div>
</div>
<!--END QUESTION.-->
<br>
<!--Q3-->
<div class="os-raise-ib-pset-problem" data-content-id="510fe714-b9ad-405d-9b78-8ed6968e7062" data-problem-type="dropdown" data-solution="Arithmetic" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li>3, 8, 13, 18</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Arithmetic </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Arithmetic. </p>
</div>
</div>
<!--END QUESTION.-->
<br>
<!--Q4-->
<div class="os-raise-ib-pset-problem" data-content-id="50feab68-5dea-4c36-be37-545bdaf60041" data-problem-type="dropdown" data-solution="Arithmetic" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li>18, 13, 8, 3</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Arithmetic </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Arithmetic. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Q5-->
<div class="os-raise-ib-pset-problem" data-content-id="cf3aaa85-e458-44cf-b486-5f63ff2a8672" data-problem-type="dropdown" data-solution="Geometric" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li>18, 9, 4.5, 2.25</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Geometric </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Geometric. </p>
</div>
</div>
<!--END QUESTION.-->
<br>
<!--Q6-->
<div class="os-raise-ib-pset-problem" data-content-id="af41a1fe-53a0-42fa-a351-2dac2cdbb508" data-problem-type="dropdown" data-solution="Arithmetic" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6" type="a">
<li>18, 20, 22, 24</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Arithmetic </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Arithmetic. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Q7-->
<div class="os-raise-ib-pset-problem" data-content-id="1b175600-5dea-49f8-ac17-d905c8b82fb1" data-problem-type="dropdown" data-solution="Geometric" data-solution-options='["Arithmetic", "Geometric"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="7" type="a">
<li>3, 15, 75, 375</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Geometric </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is Geometric. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<h4>Desmos Activity:</h4>
<br>
<div class="os-raise-usermessage-link">
<p>Log into <a href="https://student.desmos.com" target="_blank">student.desmos.com</a> using the information provided by your teacher to complete the activity.</p>
</div>
<br>
<h4>Video: Learning About Recursive Definitions and Sequences</h4>
<p>Watch the following video to learn more about matching sequences to their recursive definitions.</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/1ad29298163115b9ffa0ba588ead0d01ce6e667a">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/72685ed3769bf485d9d077fb87829e1f17f43a0d" srclang="en_us">https://k12.openstax.org/contents/raise/resources/1ad29298163115b9ffa0ba588ead0d01ce6e667a
</video></div>
</div>