e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
255 lines
11 KiB
HTML
255 lines
11 KiB
HTML
<!doctype html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/sequences-sums-arithmetic.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:52 GMT -->
|
||
<head>
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Arithmetic Sequences and Sums</title>
|
||
<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents." /><!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education" />
|
||
<meta http-equiv="content-type" content="text/html; charset=utf-8" />
|
||
<meta name="viewport" content="width=device-width; initial-scale=1.0; user-scalable=true;" />
|
||
<meta name="HandheldFriendly" content="true"/>
|
||
<meta http-equiv="pics-label" content='(PICS-1.1 "http://www.classify.org/safesurf/" L gen true for "http://www.mathsisfun.com" r (SS~~000 1))' />
|
||
<link rel="stylesheet" type="text/css" href="../style3.css" />
|
||
<script src="../main3.js" type="text/javascript"></script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
<div class="bg">
|
||
<div id="stt"></div>
|
||
<div id="hdr"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun" /></a></div>
|
||
<div id="advText">Advanced</div>
|
||
<div id="gtran"><script type="text/javascript">document.write(getTrans());</script></div>
|
||
<div id="gplus"><script type="text/javascript">document.write(getGPlus());</script></div>
|
||
<div id="adTopOuter" class="centerfull noprint">
|
||
<div id="adTop">
|
||
<script type="text/javascript">document.write(getAdTop());</script>
|
||
</div>
|
||
</div>
|
||
<div id="adHide">
|
||
<div id="showAds1"><a href="javascript:showAds()">Show Ads</a></div>
|
||
<div id="hideAds1"><a href="javascript:hideAds()">Hide Ads</a><br>
|
||
<a href="../about-ads.html">About Ads</a></div>
|
||
</div>
|
||
<div id="menuWide" class="menu">
|
||
<script type="text/javascript">document.write(getMenu(0));</script>
|
||
</div>
|
||
<div id="linkto">
|
||
<div id="linktort"><script type="text/javascript">document.write(getLinks());</script></div>
|
||
</div>
|
||
<div id="search" role="search"><script type="text/javascript">document.write(getSearch());</script></div>
|
||
<div id="menuSlim" class="menu">
|
||
<script type="text/javascript">document.write(getMenu(1));</script>
|
||
</div>
|
||
<div id="menuTiny" class="menu">
|
||
<script type="text/javascript">document.write(getMenu(2));</script>
|
||
</div>
|
||
<div id="extra"></div>
|
||
</div>
|
||
<div id="content" role="main"><!-- #BeginEditable "Body" -->
|
||
|
||
<h1 align="center">Arithmetic Sequences and Sums</h1>
|
||
<h2>Sequence</h2>
|
||
<p>A <a href="sequences-series.html">Sequence</a> is a set of things (usually numbers) that are in order. </p>
|
||
<p class="center"><img src="images/sequence.svg" alt="Sequence" /></p>
|
||
<p>Each number in the sequence is called a <b>term</b> (or sometimes "element" or "member"), read <a href="sequences-series.html">Sequences and Series</a> for more details.</p>
|
||
<h2>Arithmetic Sequence</h2>
|
||
<p>In an Arithmetic Sequence <b>the difference between one term and the next is a constant</b>.</p>
|
||
<p>In other words, we just add the same value each time ... infinitely.</p>
|
||
<div class="example">
|
||
<h3>Example:</h3>
|
||
<div class="simple">
|
||
<table align="center">
|
||
<tr>
|
||
<td><font size="+1" class="large">1, 4, 7, 10, 13, 16, 19, 22, 25, ...</font></td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
<p>This sequence has a difference of 3 between each number. <br />
|
||
The pattern is continued by <b>adding 3</b> to the last number each time, like this:</p>
|
||
<p class="center"><img src="images/arithmetic-sequence-1-3.svg" alt="arithmetic sequence 1,4,7,10," /></p>
|
||
</div>
|
||
<p><b>In General</b> we could write an arithmetic sequence like this:</p>
|
||
<p align="center" class="large">{a, a+d, a+2d, a+3d, ... }</p>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li><b>a</b> is the first term, and </li>
|
||
<li><b>d</b> is the difference between the terms (called the <b>"common difference"</b>)</li>
|
||
</ul><p> </p>
|
||
<div class="example">
|
||
<h3>Example: (continued)</h3>
|
||
<div class="simple">
|
||
<table align="center">
|
||
<tr>
|
||
<td><font size="+1" class="large">1, 4, 7, 10, 13, 16, 19, 22, 25, ...</font></td>
|
||
</tr>
|
||
</table></div>
|
||
<p>Has:</p>
|
||
<ul>
|
||
<li>a = 1 (the first term)</li>
|
||
<li>d = 3 (the "common difference" between terms)</li>
|
||
</ul>
|
||
<p>And we get:</p>
|
||
<p class="center larger">{a, a+d, a+2d, a+3d, ... }</p>
|
||
<p class="center larger">{1, 1+3, 1+2×3, 1+3×3, ... }</p>
|
||
<p class="center larger">{1, 4, 7, 10, ... }</p>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Rule</h3>
|
||
<p>We can write an Arithmetic Sequence as a rule:</p>
|
||
<p align="center"> <span class="large">x<sub>n</sub> = a + d(n−1)</span></p>
|
||
<p align="center">(We use "n−1" because <b>d</b> is not used in the 1st term).<br />
|
||
</p>
|
||
<div class="example">
|
||
<h3>Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:</h3>
|
||
<div class="simple">
|
||
<table align="center">
|
||
<tr>
|
||
<td><font size="+1">3, 8, 13, 18, 23, 28, 33, 38,</font><font size="+1" class="large"> ...</font></td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
<p>This sequence has a difference of 5 between each number.</p>
|
||
<p class="center"><img src="images/arithmetic-sequence-3-5.svg" alt="arithmetic sequence 3,8,13,18" /></p>
|
||
<p>The values of <b>a</b> and <b>d</b> are:</p>
|
||
<ul>
|
||
<li><b>a = 3</b> (the first term)</li>
|
||
<li><b>d = 5</b>
|
||
(the "common difference")</li>
|
||
</ul>
|
||
<p>Using the Arithmetic Sequence rule:</p>
|
||
<p align="center" class="larger"><b>x<sub>n</sub></b> = a + d(n−1)</p>
|
||
<p align="center" class="larger">= 3 + 5(n−1) </p>
|
||
<p align="center" class="larger">= 3 + 5n − 5 </p>
|
||
<p align="center" class="larger">= <b>5n − 2</b></p>
|
||
<p>So the 9th term is:</p>
|
||
<p align="center" class="larger">x<sub>9</sub> = 5×9 − 2 <br>
|
||
= 43</p>
|
||
<p>Is that right? Check for yourself!</p>
|
||
</div><p> </p>
|
||
<div class="words">
|
||
<p>Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s)</p>
|
||
</div>
|
||
<h2>Advanced Topic: Summing an Arithmetic Series</h2>
|
||
<p><b>To sum up</b> the terms of this arithmetic sequence:</p>
|
||
<p align="center"><span class="large">a + (a+d) + (a+2d) + (a+3d) + ... </span></p>
|
||
<p> use this formula:</p>
|
||
<p align="center"><img src="images/partial-sum-h.gif" alt="Sigma" width="274" height="57" /></p>
|
||
<div class="center80">
|
||
<p>What is that funny symbol? It is called <a href="sigma-notation.html">Sigma Notation</a></p>
|
||
<table border="0" align="center">
|
||
<tr>
|
||
<td><img src="images/sigma.gif" alt="Sigma" width="32" height="34" /></td>
|
||
<td> (called Sigma) means "sum up"</td>
|
||
</tr>
|
||
</table>
|
||
<p>And below and above it are shown the starting and ending values:</p>
|
||
<p align="center"><img src="images/sigma-notation.svg" alt="Sigma Notation" /></p>
|
||
<p align="center">It says "Sum up <i><b>n</b></i> where <i><b>n</b></i> goes from 1 to 4. Answer=<b>10</b></p>
|
||
</div>
|
||
<p>Here is how to use it:</p>
|
||
<div class="example">
|
||
<h3>Example: Add up the first 10 terms of the arithmetic sequence:</h3>
|
||
<p align="center" class="large"> { 1, 4, 7, 10, 13, ... }</p>
|
||
<p>The values of <b>a</b>, <b>d</b> and <b>n</b> are:</p>
|
||
<ul>
|
||
<li><b>a = 1</b> (the first term)</li>
|
||
<li><b>d = 3</b> (the "common difference" between terms)</li>
|
||
<li><b>n = 10</b> (how many terms to add up)</li>
|
||
</ul>
|
||
|
||
<p>So:</p>
|
||
<p align="center"><img src="images/partial-sum-h.gif" alt="Sigma" width="274" height="57" /></p>
|
||
<p>Becomes:</p>
|
||
<p align="center"><img src="images/partial-sum-h2.gif" alt="Sigma" width="338" height="60" /></p>
|
||
<p align="center" class="large">= 5(2+9·3) = 5(29) = 145</p>
|
||
<p> </p>
|
||
<p><i>Check: why don't you add up the terms yourself, and see if it comes to 145</i></p>
|
||
</div>
|
||
<h2>Footnote: Why Does the Formula Work?</h2>
|
||
<p>Let's see <b>why</b> the formula works, because we get to use an interesting "trick" which is worth knowing.</p>
|
||
<p><b>First</b>, we will call the whole sum <b>"S"</b>:</p>
|
||
<div class="so"> S = a + (a + d) + ... + (a + (n−2)d) + (a + (n−1)d) </div>
|
||
<p><b>Next</b>, rewrite S in reverse order:</p>
|
||
<div class="so"> S = (a + (n−1)d) + (a + (n−2)d) + ... + (a + d) + a</div>
|
||
<p>Now add those two, term by term:</p>
|
||
<table border="0" align="center" cellpadding="4">
|
||
<tr style="white-space:nowrap;">
|
||
<td>S</td>
|
||
<td>=</td>
|
||
<td>a</td>
|
||
<td>+</td>
|
||
<td>(a+d)</td>
|
||
<td>+</td>
|
||
<td>...</td>
|
||
<td>+</td>
|
||
<td>(a + (n-2)d)</td>
|
||
<td>+</td>
|
||
<td>(a + (n-1)d)</td>
|
||
</tr>
|
||
<tr style="white-space:nowrap;">
|
||
<td>S</td>
|
||
<td>=</td>
|
||
<td>(a + (n-1)d)</td>
|
||
<td>+</td>
|
||
<td>(a + (n-2)d)</td>
|
||
<td>+</td>
|
||
<td>...</td>
|
||
<td>+</td>
|
||
<td>(a + d)</td>
|
||
<td>+</td>
|
||
<td>a</td>
|
||
</tr>
|
||
<tr>
|
||
<td colspan="11" style="border-top: 1px solid black; height:1px;"></td>
|
||
</tr>
|
||
<tr style="white-space:nowrap; border-top: 1px solid black;">
|
||
<td>2S</td>
|
||
<td>=</td>
|
||
<td>(2a + (n-1)d)</td>
|
||
<td>+</td>
|
||
<td>(2a + (n-1)d)</td>
|
||
<td>+</td>
|
||
<td>...</td>
|
||
<td>+</td>
|
||
<td>(2a + (n-1)d)</td>
|
||
<td>+</td>
|
||
<td>(2a + (n-1)d)</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<p><b>Each term is the same! </b>And there are "n" of them so ...</p>
|
||
<div class="so"> 2S = n × (2a + (n−1)d) </div>
|
||
<p>Now, just divide by 2 and we get:</p>
|
||
<div class="center80">
|
||
<p align="center" class="large">S = (n/2) × (2a + (n−1)d) </p>
|
||
</div>
|
||
<p>Which is our formula:</p>
|
||
<p align="center"><img src="images/partial-sum-h.gif" alt="Sigma" width="274" height="57" /></p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
|
||
<script type="text/javascript">getQ(8291, 8309, 604, 605, 606, 607, 1252, 1253, 1254, 1251);</script>
|
||
|
||
</div>
|
||
|
||
<div class="related"><a href="sequences-series.html">Sequences</a> <a href="sequences-sums-geometric.html">Geometric Sequences and Sums</a> <a href="partial-sums.html">Partial Sums</a> <a href="sigma-notation.html">Sigma Notation</a> <a href="index.html">Algebra Menu</a></div>
|
||
<!-- #EndEditable --></div>
|
||
<div id="adend" class="centerfull noprint">
|
||
<script type="text/javascript">document.write(getAdEnd());</script>
|
||
</div>
|
||
<div id="footer" class="centerfull noprint">
|
||
<script type="text/javascript">document.write(getFooter());</script>
|
||
</div>
|
||
<div id="copyrt">
|
||
Copyright © 2017 MathsIsFun.com
|
||
</div>
|
||
|
||
<script type="text/javascript">document.write(getBodyEnd());</script>
|
||
</body>
|
||
<!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/sequences-sums-arithmetic.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:53 GMT -->
|
||
</html>
|