1<!DOCTYPE html>
2<html>
3 <head>
4 <meta charset="utf-8" />
5 <title>sequences</title><script type="text/x-mathjax-config">
6 MathJax.Hub.Config({
7 jax: ["input/TeX","output/HTML-CSS"],
8 extensions: ["[a11y]/accessibility-menu.js"],
9 TeX: {
10 "Macros": {},
11 "equationNumbers": {
12 "autoNumber": "AMS",
13 "useLabelIds": false
14 },
15 "extensions": [
16 "AMSmath.js",
17 "AMSsymbols.js",
18 "noErrors.js",
19 "noUndefined.js"
20 ]
21},
22 showMathMenu: true
23 });
24</script>
25<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js"></script>
26 <style>body { padding: 0; margin: 0; }
27.markdown-preview-plus-view:not([data-use-github-style]) {
28 font-size: 1em;
29 font-family: Helvetica;
30 column-count: 2;
31}
32.markdown-preview-plus-view:not([data-use-github-style]) > :first-child {
33 margin-top: 0;
34}
35.markdown-preview-plus-view:not([data-use-github-style]) h1,
36.markdown-preview-plus-view:not([data-use-github-style]) h2,
37.markdown-preview-plus-view:not([data-use-github-style]) h3,
38.markdown-preview-plus-view:not([data-use-github-style]) h4,
39.markdown-preview-plus-view:not([data-use-github-style]) h5,
40.markdown-preview-plus-view:not([data-use-github-style]) h6 {
41 line-height: 1.2;
42 /* margin-top: 1.5em; */
43 margin-bottom: 0.5em;
44}
45}
46</style>
47 </head>
49 <body class="markdown-preview-plus-view">
50 <h1>Sequences and Series</h1>
51<p><span class="math"><script type="math/tex">\{\ a,\ b,\ c,\ d\ ...\ \}</script></span></p>
52<ul>
53<li><span class="math"><script type="math/tex">n^{th}</script></span> iteration: <span class="math"><script type="math/tex">t_n</script></span></li>
54<li><span class="math"><script type="math/tex">n</script></span> can start at <span class="math"><script type="math/tex">0</script></span> or <span class="math"><script type="math/tex">1</script></span></li>
55</ul>
56<h2>Defining sequences</h2>
57<ul>
58<li>
59<p>sequence rule in terms of <span class="math"><script type="math/tex">n</script></span> e.g. <span class="math"><script type="math/tex">t_n = 2n</script></span></p>
60</li>
61<li>
62<p>recurrence relation - value of term is derived from previous term (recursion)<br>
63e.g. <span class="math"><script type="math/tex">\ t_n=t_{n-1}+7,\ t_1=4</script></span></p>
64<ul>
65<li><span class="math"><script type="math/tex">t_1=4</script></span></li>
66<li><span class="math"><script type="math/tex">t_2=4+7=11</script></span></li>
67<li><span class="math"><script type="math/tex">t_3=11+7=17</script></span></li>
68<li><span class="math"><script type="math/tex">\dots</script></span></li>
69</ul>
70</li>
71</ul>
72<h2>Arithmetic sequences</h2>
73<p><span class="math"><script type="math/tex">t_n = a+(n-1)d</script></span></p>
74<p>where<br>
75<span class="math"><script type="math/tex">a=</script></span> first term<br>
76<span class="math"><script type="math/tex">d=</script></span> common difference</p>
77<h3>Arithmetic mean</h3>
78<p><span class="math"><script type="math/tex">m_a = \frac{a+b}{2}</script></span></p>
79<h3>Arithmetic series</h3>
80<p><span class="math"><script type="math/tex">\sum</script></span> of all terms in an arithmetic sequence.</p>
81<p>e.g. A sequence is defined by <span class="math"><script type="math/tex">t_n = a+(n-1)d</script></span></p>
82<p><span class="math"><script type="math/tex">s_n = {\frac{n}{2}}(2a+(n-1)d)</script></span> <–sum of <span class="math"><script type="math/tex">n</script></span> terms, including <span class="math"><script type="math/tex">t_n</script></span> (works only with <span class="math"><script type="math/tex">t_1</script></span>)<br>
83where</p>
84<ul>
85<li><span class="math"><script type="math/tex">a=t_1</script></span> (first term)</li>
86<li><span class="math"><script type="math/tex">l=t_n</script></span> (last term)</li>
87</ul>
88<h2>Geometric sequences</h2>
89<p><span class="math"><script type="math/tex">t_n=ar^{n-1}</script></span><br>
90where<br>
91<span class="math"><script type="math/tex">a =</script></span> first term<br>
92<span class="math"><script type="math/tex">r =</script></span> common ratio of successive terms (<span class="math"><script type="math/tex">r=\frac{t_k}{t_{k-1}}</script></span>)</p>
93<h3>Geometric mean</h3>
94<p><span class="math"><script type="math/tex">m_g</script></span> of <span class="math"><script type="math/tex">a</script></span> and <span class="math"><script type="math/tex">b</script></span> is <span class="math"><script type="math/tex">\sqrt{ab}</script></span></p>
95<p>If <span class="math"><script type="math/tex">a, c, b</script></span> are positive and consecutive terms in a geometric sequence, then:<br>
96<span class="math"><script type="math/tex">{\frac{c}{a}}={\frac{b}{c}}\therefore c=\sqrt{ab}</script></span>,</p>
97<h3>Geometric series</h3>
98<p><span class="math"><script type="math/tex">\sum</script></span> of all terms in a geometric sequence.</p>
99<p>e.g. <span class="math"><script type="math/tex">s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}</script></span></p>
100<p><span class="math"><script type="math/tex">rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}</script></span></p>
101<p><span class="math"><script type="math/tex">rs_n - s_n = -a + ar^n</script></span></p>
102<hr>
103<p><span class="math"><script type="math/tex">s_n = {\frac{a(r^n-1)}{r-1}}</script></span><br>
104or<br>
105<span class="math"><script type="math/tex">s_n = {\frac{a(r^n-1)}{r-1}}</script></span></p>
106<h2>Infinite series</h2>
107<p>If <span class="math"><script type="math/tex">-1 < r -1</script></span>, the infinite geometric series <span class="math"><script type="math/tex">a+ar+ar^2 \dots</script></span> is convergent.</p>
108<p>Sum to infinity is given by</p>
109<p><span class="math"><script type="math/tex">s_\infty={\frac{a}{1-r}}</script></span></p>
110<p>Tennis ball question - remember down <strong>and</strong> up strokes. Multiply down strokes by 2, subtract 1.</p>
111 </body>
113</html>