spec / sequences.htmlon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (06ac421)
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  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> &lt;–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>
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