4a5382b5-b3ed-44f1-a179-331187658026.html

<h4>Writing Expressions for Patterns</h4>
<p class="os-raise-text-bold">Example 1</p>
<p><img alt="DIAGRAM WITH 3 FIGURES. STEP 1 IS SHAPED LIKE A PLUS SIGN, WITH 1 SQUARE IN THE MIDDLE AND 1 SQUARE ATTACHED TO EACH OF ITS SIDES. STEP 2 HAS A RECTANGLE WITH 2 ROWS AND 4 COLUMNS. ABOVE AND BELOW THE SECOND COLUMN OF THE RECTANGLE IS 1 SQUARE. STEP 3 HAS A RECTANGLE WITH 3 ROWS AND 5 COLUMNS. ABOVE AND BELOW THE SECOND COLUMN OF THE RECTANGLE IS 1 SQUARE.
" height="238" src="https://k12.openstax.org/contents/raise/resources/0a1a836e44bf7cd832a0d520642fc03dad9e66c3"
    width="550"></p>
<p>Find the number of squares in Step 5:</p>
<p>First, look to see the pattern. Every step, another row and another column are added, plus the two extra squares.</p>
<ul>
  <li> Step 1: 1 row of \(3 + 2\) </li>
  <li> Step 2: 2 rows of \(4 +2\) </li>
  <li> Step 3: 3 rows of \(5 + 2\) </li>
</ul>
<p>So the width is always 1 by \(n\), or \(n\). The length is \(n+2\), and then another 2 is added.</p>
<p>Then, draw pictures or create a table:</p>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">
        Step</strong>
      </th>
      <th scope="col">
        # of squares</strong>
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <p>1
      </td>
      <td>
        <p>5
      </td>
    </tr>
    <tr>
      <td>
        <p>2
      </td>
      <td>
        <p>10
      </td>
    </tr>
    <tr>
      <td>
        <p>3
      </td>
      <td>
        <p>17
      </td>
    </tr>
    <tr>
      <td>
        <p>4
      </td>
      <td>
        <p>26
      </td>
    </tr>
    <tr>
      <td>
        <p>5
      </td>
      <td>
        <p>37
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>At Step 5, there are 37 squares.</p>
<p class="os-raise-text-bold">Example 2</p>
<p>Find the expression for Step \(n\).</p>
<p>Multiply the length and width, and then add the 2 extra squares:</p>
<p>\(n(n+2)+2 = n^2+2n+n\)</p>
<h4>Try It: Writing Expressions for Patterns</h4>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
    <p>Use the figure below to find the number of squares in Step 5 and then Step \(n\).</p>
    <p><img alt="DIAGRAM WITH 3 FIGURES. STEP 1 HAS A RECTANGLE WITH 2 ROWS AND 3 COLUMNS. 2 SQUARES OF A THIRD ROW ARE ADDED. STEP 2 HAS A RECTANGLE WITH 3 ROWS AND 4 COLUMNS. 3 SQUARES OF A FOURTH ROW ARE ADDED. STEP 3 HAS A RECTANGLE WITH 4 ROWS AND 5 COLUMNS. 4 SQUARES OF A FIFTH ROW ARE ADDED.
" height="175" src="https://k12.openstax.org/contents/raise/resources/1b6a05a9afdaeddde371b9f9242292998e4cc777"
        width="410"></p>
  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
  </div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
  <p>Here is how to determine the number of squares in each step:</p>
  <p>Notice that each figure adds a row and column, and then subtracts one square. The number of rows is always 2 more
    than the step number.</p>
  <p>Begin by making a table:</p>
  <table class="os-raise-midsizetable">
    <thead>
      <tr>
        <th scope="col">
          Step
        </th>
        <th scope="col">
          # of squares
        </th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          1
        </td>
        <td>
          8
        </td>
      </tr>
      <tr>
        <td>
          2
        </td>
        <td>
          15
        </td>
      </tr>
      <tr>
        <td>
          3
        </td>
        <td>
          24
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          35
        </td>
      </tr>
      <tr>
        <td>
          5
        </td>
        <td>
          48
        </td>
      </tr>
    </tbody>
  </table>
  <br>
  <p>There are 48 squares in Step 5.</p>
  <p>Step \(n\) would be found with \((n+2)(n+2)-1\) or \((n+2)^2-1\).</p>
</div>