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533 lines
22 KiB
HTML
533 lines
22 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Sequences</h1>
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<p class="center"><i>You can read a gentle introduction to Sequences in <a href="../numberpatterns.html">Common Number Patterns</a>.</i></p>
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<h2>What is a Sequence?</h2>
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<div class="center80">
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<p class="center larger">A Sequence is a list of things (usually numbers) that are in order.</p>
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</div>
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<p class="center"><img src="images/sequence.svg" alt="Sequence 3,5,7,9,..." style="max-width:100%" height="168" width="422"></p>
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<h2>Infinite or Finite</h2>
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<p class="center larger">When the sequence goes on forever it is called an <b>infinite sequence</b>,<br>
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otherwise it is a <b>finite sequence</b></p>
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<div class="example">
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<h3>Examples:</h3>
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<p>{1, 2, 3, 4, ...} is a very simple sequence (and it is an <b>infinite sequence</b>)</p>
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<p>{20, 25, 30, 35, ...} is also an infinite sequence</p>
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<p>{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a <b>finite sequence</b>)</p>
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<p>{4, 3, 2, 1} is 4 to 1 <b>backwards</b></p>
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<p>{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles</p>
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<p>{a, b, c, d, e} is the sequence of the first 5 letters <b>alphabetically</b></p>
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<p>{f, r, e, d} is the sequence of letters in the name "fred"</p>
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<p>{0, 1, 0, 1, 0, 1, ...} is the sequence of <b>alternating</b> 0s and 1s (yes they are in order, it is an alternating order in this case)</p>
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</div>
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<h2>In Order</h2>
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<p>When we say the terms are "in order", we are free to define <b>what order that is</b>! They could go forwards, backwards ... or they could alternate ... or any type of order we want!</p>
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<h2>Like a Set</h2>
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<p>A Sequence is like a <a href="../sets/sets-introduction.html">Set</a>, except:</p>
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<ul>
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<li>the terms are <b>in order</b> (with Sets the order does not matter)</li>
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<li>the same value can appear many times (only once in Sets)</li>
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</ul>
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<div class="example">
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<p>Example: {0, 1, 0, 1, 0, 1, ...} is the <b>sequence</b> of alternating 0s and 1s.</p>
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<p>The <b>set</b> is just {0,1}</p>
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</div>
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<h2>Notation</h2>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;">Sequences also use the same <b>notation</b> as sets:<br>
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list each element, separated by a comma,<br>
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and then put curly brackets around the whole thing.</td>
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<td class="larger" align="center" width="200">{3, 5, 7, ...}</td>
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</tr>
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</tbody></table>
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<div class="def">
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<p class="center">The curly brackets <span class="large">{ }</span> are sometimes called "set brackets" or "braces".</p>
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</div>
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<h2>A Rule</h2>
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<p>A Sequence usually has a <b>Rule</b>, which is a way to find the value of each term.</p>
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<div class="example">
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<p>Example: the sequence {3, 5, 7, 9, ...} starts at 3 and jumps 2 every time:</p>
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<p class="center"><img src="images/sequence2.gif" alt="{3, 5, 7, 9, ...}" height="102" width="347"></p>
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</div>
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<h2>As a Formula</h2>
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<p>Saying "<i>starts at 3 and jumps 2 every time</i>" is fine, but it doesn't help us calculate the:</p>
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<ul>
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<li>10<sup>th</sup> term,</li>
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<li>100<sup>th</sup> term, or</li>
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<li><i><b>n</b></i><sup>th</sup> term, where <i><b>n</b></i> could be any term number we want.</li>
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</ul>
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<div class="center80">
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<p class="center large">So, we want a formula with "<span class="larger">n</span>" in it (where <span class="larger">n</span> is any term number).</p>
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</div>
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<h3>So, What Can A Rule For {3, 5, 7, 9, ...} Be?</h3>
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<p>Firstly, we can see the sequence goes up 2 every time, so we can <b>guess</b> that a Rule is something like "2 times n" (where "n" is the term number). Let's test it out:</p>
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<p class="center large">Test Rule: 2n</p>
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<div class="beach">
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<table align="center" width="50%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th>n</th>
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<th>Term</th>
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<th>Test Rule</th>
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</tr>
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<tr style="text-align:center;">
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<td><b>1</b></td>
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<td>3</td>
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<td>2<b>n</b> = 2×<b>1</b> = <b>2</b></td>
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</tr>
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<tr style="text-align:center;">
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<td><b>2</b></td>
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<td>5</td>
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<td>2<b>n</b> = 2×<b>2</b> = <b>4</b></td>
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</tr>
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<tr style="text-align:center;">
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<td><b>3</b></td>
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<td>7</td>
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<td>2<b>n</b> = 2×<b>3</b> = <b>6</b></td>
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</tr>
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</tbody></table>
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</div>
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<p>That <b>nearly</b> worked ... but it is <b>too low</b> by 1 every time, so let us try changing it to:</p>
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<p class="center large">Test Rule: 2n+1</p>
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<div class="beach">
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<table align="center" width="50%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th>n</th>
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<th>Term</th>
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<th> Test Rule</th>
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</tr>
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<tr style="text-align:center;">
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<td><b>1</b></td>
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<td>3</td>
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<td>2<b>n</b>+1 = 2×<b>1</b> + 1 = 3</td>
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</tr>
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<tr style="text-align:center;">
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<td><b>2</b></td>
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<td>5</td>
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<td>2<b>n</b>+1 = 2×<b>2</b> + 1 = 5</td>
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</tr>
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<tr style="text-align:center;">
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<td><b>3</b></td>
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<td>7</td>
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<td>2<b>n</b>+1 = 2×<b>3</b> + 1 = 7</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"><b>That Works!</b></p>
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<p>So instead of saying "starts at 3 and jumps 2 every time" we write this:</p>
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<p class="center larger"><b>2n+1</b></p>
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<div class="example">
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<p>Now we can calculate, for example, the <b>100th term</b>:</p>
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<p class="center">2 × 100 + 1 = <b>201</b></p>
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</div>
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<h2>Many Rules</h2>
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<p>But mathematics is so powerful we can find <b>more than one Rule</b> that works for any sequence.</p>
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<div class="example">
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<h3>Example: the sequence {3, 5, 7, 9, ...}</h3>
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<p>We have just shown a Rule for {3, 5, 7, 9, ...} is: <b>2n+1</b></p>
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<p class="center">And so we get: {3, 5, 7, 9, 11, 13, ...}</p>
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<p>But can we find another rule?</p>
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<p>How about <b>"odd numbers without a 1 in them"</b>:</p>
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<p class="center">And we get: {3, 5, 7, 9, 23, 25, ...}</p>
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<p><b>A completely different sequence!</b></p>
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<p>And we could find more rules that match <b>{3, 5, 7, 9, ...}</b>. Really we could.</p>
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</div>
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<p>So it is best to say "A Rule" rather than "The Rule" (unless we know it is the right Rule).</p>
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<h2>Notation</h2>
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<p>To make it easier to use rules, we often use this special style:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/sequence-term.svg" alt="sequence term" height="85" width="218"></td>
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<td>
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<ul>
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<li class="larger"><b>x<sub>n</sub></b> is the term</li>
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<li class="larger"><b>n</b> is the term number</li>
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</ul></td>
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</tr>
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</tbody></table><br>
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</div>
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<div class="example">
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<p>Example: to mention the "5th term" we write: <span class="larger">x<sub>5</sub></span></p>
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</div>
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<p>So a rule for <b>{3, 5, 7, 9, ...}</b> can be written as an equation like this:</p>
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<p class="center large">x<sub>n</sub> = 2n+1</p>
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<p>And to calculate the 10th term we can write:</p>
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<p class="center larger">x<sub>10</sub> = 2<b>n</b>+1 = 2×<b>10</b>+1 = 21</p>
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<p class="center"><i>Can you calculate <span class="larger">x<sub>50</sub></span> (the 50th term) doing this?</i></p>
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<p>Here is another example:</p>
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<div class="example">
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<h3>Example: Calculate the first 4 terms of this sequence:</h3>
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<h3 align="center">{a<sub>n</sub>} = { (-1/n)<sup>n</sup> }</h3>
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<p>Calculations:</p>
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<ul>
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<li>a<sub>1</sub> = (-1/1)<sup>1</sup> = -1</li>
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<li>a<sub>2</sub> = (-1/2)<sup>2</sup> = 1/4</li>
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<li>a<sub>3</sub> = (-1/3)<sup>3</sup> = -1/27</li>
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<li>a<sub>4</sub> = (-1/4)<sup>4</sup> = 1/256</li>
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</ul>
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<p>Answer:</p>
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<p class="center larger">{a<sub>n</sub>} = { -1, 1/4, -1/27, 1/256, ... }</p>
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</div>
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<h2>Special Sequences</h2>
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<p>Now let's look at some special sequences, and their rules.</p>
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<h2>Arithmetic Sequences</h2>
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<p>In an <a href="sequences-sums-arithmetic.html">Arithmetic Sequence</a> <b>the difference between one term and the next is a constant</b>.</p>
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<p>In other words, we just add some value each time ... on to infinity.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td><font class="large" size="+1">1, 4, 7, 10, 13, 16, 19, 22, 25, ...</font></td>
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</tr>
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</tbody></table>
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</div>
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<div class="center">
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<p>This sequence has a difference of 3 between each number.<br>
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Its Rule is <b>x<sub>n</sub> = 3n-2</b></p>
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</div>
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</div>
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<p><b>In General</b> we can write an arithmetic sequence like this:</p>
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<p class="center large">{a, a+d, a+2d, a+3d, ... }</p>
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<p>where:</p>
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<ul>
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<li><b>a</b> is the first term, and</li>
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<li><b>d</b> is the difference between the terms (called the <b>"common difference"</b>)</li>
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</ul>
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<p>And we can make the rule:</p>
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<p class="center"><span class="large">x<sub>n</sub> = a + d(n-1)</span></p>
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<p class="center">(We use "n-1" because <b>d</b> is not used in the 1st term).</p>
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<h2>Geometric Sequences</h2>
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<p>In a <a href="sequences-sums-geometric.html">Geometric Sequence</a> each term is found by <b>multiplying</b> the previous term by a <b>constant</b>.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td><font size="+1">2, 4, 8, 16, 32, 64, 128, 256</font><font class="large" size="+1">, ...</font></td>
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</tr>
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</tbody></table>
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</div>
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<p class="center">This sequence has a factor of 2 between each number.<br>
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Its Rule is <b>x<sub>n</sub> = 2<sup>n</sup></b></p>
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</div>
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<p><b>In General</b> we can write a geometric sequence like this:</p>
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<p class="center large">{a, ar, ar<sup>2</sup>, ar<sup>3</sup>, ... }</p>
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<p>where:</p>
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<ul>
|
||
<li><b>a</b> is the first term, and</li>
|
||
<li><b>r</b> is the factor between the terms (called the <b>"common ratio"</b>)</li>
|
||
</ul>
|
||
<div class="center80">
|
||
<p>Note: <b>r</b> should not be 0.</p>
|
||
<ul>
|
||
<li>When <b>r=0</b>, we get the sequence {a,0,0,...} which is not geometric</li>
|
||
</ul>
|
||
</div>
|
||
<p>And the rule is:</p>
|
||
<p class="center large">x<sub>n</sub> = ar<sup>(n-1)</sup></p>
|
||
<p class="center">(We use "n-1" because <span class="large">ar<sup>0</sup></span> is the 1st term)</p>
|
||
|
||
|
||
<h2><a name="triangular" id="triangular"></a>Triangular Numbers</h2>
|
||
|
||
<div class="simple">
|
||
|
||
<table align="center">
|
||
<tbody>
|
||
<tr>
|
||
<td><font size="+1">1, 3, 6, 10, 15, 21, 28, 36, 45</font><font class="large" size="+1">, ...</font></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>The <a href="triangular-numbers.html">Triangular Number Sequence</a> is generated from a pattern of dots which form a
|
||
triangle:</p>
|
||
<p class="center"><img src="../numbers/images/triangular-number-dots.svg" alt="triangular numbers" style="max-width:100%" height="177" width="473"></p>
|
||
<p>By adding another row of dots and counting all the dots we can find
|
||
the next number of the sequence.</p>
|
||
<center>
|
||
<p align="left">But it is easier to use this Rule:</p>
|
||
<p class="large">x<sub>n</sub> = n(n+1)/2</p>
|
||
</center>
|
||
|
||
<div class="example">
|
||
<p>Example:</p>
|
||
<ul>
|
||
<li>the 5th Triangular Number is x<sub>5</sub> = 5(5+1)/2 = <b>15</b>,</li>
|
||
<li>and the sixth is x<sub>6</sub> = 6(6+1)/2 = <b>21</b></li>
|
||
</ul>
|
||
</div>
|
||
|
||
|
||
<h2>Square Numbers</h2>
|
||
|
||
<div class="simple">
|
||
|
||
<table align="center">
|
||
<tbody>
|
||
<tr>
|
||
<td><font size="+1">1, 4, 9, 16, 25, 36, 49, 64, 81</font><font class="large" size="+1">, ...</font></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>The next number is made by squaring where it is in the pattern.</p>
|
||
<p class="center larger">Rule is<b> x<sub>n</sub> = n<sup>2</sup></b></p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Cube Numbers</h2>
|
||
|
||
<div class="simple">
|
||
|
||
<table align="center">
|
||
<tbody>
|
||
<tr>
|
||
<td><font size="+1">1, 8, 27, 64, 125, 216, 343, 512, 729</font><font class="large" size="+1">, ...</font></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>The next number is made by cubing where it is in the pattern.</p>
|
||
<p class="center larger">Rule is<b> x<sub>n</sub> = n<sup>3</sup></b></p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Fibonacci Sequence</h2>
|
||
|
||
<div class="simple">
|
||
<p>This is the <a href="../numbers/fibonacci-sequence.html">Fibonacci Sequence</a></p>
|
||
|
||
<table align="center">
|
||
<tbody>
|
||
<tr>
|
||
<td><font size="+1">0, 1, 1, 2, 3, 5, 8, 13, 21, 34</font><font class="large" size="+1">, ...</font></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>The next number is found by <b>adding the two numbers before it</b> together:</p>
|
||
<ul>
|
||
<li>The 2 is found by adding the two numbers before it (1+1)</li>
|
||
<li>The 21 is found by adding the two numbers before it (8+13)</li>
|
||
<li>etc...</li>
|
||
</ul>
|
||
<p class="center larger">Rule is <b> x<sub>n</sub> = x<sub>n-1</sub> + x<sub>n-2</sub></b></p>
|
||
|
||
|
||
<p>That rule is interesting because it depends on the values of the previous two terms.</p>
|
||
<div class="words">
|
||
<p>Rules like that are called <b>recursive</b> formulas.</p>
|
||
</div>
|
||
<p>The Fibonacci Sequence is numbered <b>from 0 onwards</b> like this:</p>
|
||
<div class="beach">
|
||
|
||
<table align="center" cellspacing="0" cellpadding="0">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:center; width:60px;"><i>n =</i></td>
|
||
<td style="text-align:center; width:35px;"><i>0</i></td>
|
||
<td style="text-align:center; width:35px;"><i>1</i></td>
|
||
<td style="text-align:center; width:35px;"><i>2</i></td>
|
||
<td style="text-align:center; width:35px;"><i>3</i></td>
|
||
<td style="text-align:center; width:35px;"><i>4</i></td>
|
||
<td style="text-align:center; width:35px;"><i>5</i></td>
|
||
<td style="text-align:center; width:35px;"><i>6</i></td>
|
||
<td style="text-align:center; width:35px;"><i>7</i></td>
|
||
<td style="text-align:center; width:35px;"><i>8</i></td>
|
||
<td style="text-align:center; width:35px;"><i>9</i></td>
|
||
<td style="text-align:center; width:35px;"><i>10</i></td>
|
||
<td style="text-align:center; width:35px;"><i>11</i></td>
|
||
<td style="text-align:center; width:35px;"><i>12</i></td>
|
||
<td style="text-align:center; width:35px;"><i>13</i></td>
|
||
<td style="text-align:center; width:35px;"><i>14</i></td>
|
||
<td style="text-align:center; width:35px;"><i>...</i></td>
|
||
</tr>
|
||
<tr class="large">
|
||
<td style="text-align:center; width:60px;">x<sub>n</sub> =</td>
|
||
<td style="text-align:center; width:35px;">0</td>
|
||
<td style="text-align:center; width:35px;">1</td>
|
||
<td style="text-align:center; width:35px;">1</td>
|
||
<td style="text-align:center; width:35px;">2</td>
|
||
<td style="text-align:center; width:35px;">3</td>
|
||
<td style="text-align:center; width:35px;">5</td>
|
||
<td style="text-align:center; width:35px;">8</td>
|
||
<td style="text-align:center; width:35px;">13</td>
|
||
<td style="text-align:center; width:35px;">21</td>
|
||
<td style="text-align:center; width:35px;">34</td>
|
||
<td style="text-align:center; width:35px;">55</td>
|
||
<td style="text-align:center; width:35px;">89</td>
|
||
<td style="text-align:center; width:35px;">144</td>
|
||
<td style="text-align:center; width:35px;">233</td>
|
||
<td style="text-align:center; width:35px;">377</td>
|
||
<td style="text-align:center; width:35px;">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="example">
|
||
<p>Example: term "6" is calculated like this:</p>
|
||
<p class="center"><b> x<sub>6</sub> = x<sub>6-1</sub> + x<sub>6-2</sub> = x<sub>5</sub> + x<sub>4</sub> = 5 + 3 = 8</b></p>
|
||
</div>
|
||
|
||
|
||
<h2>Series and Partial Sums</h2>
|
||
|
||
<p>Now you know about sequences, the next thing to learn about is how to sum them up. Read our page on <a href="partial-sums.html">Partial Sums</a>.</p>
|
||
<div class="words">
|
||
<p>When we <b>sum</b> up just <b>part</b> of a sequence it is called a <a href="partial-sums.html">Partial Sum</a>.</p>
|
||
<p>But a <b>sum</b> of an <b>infinite</b> sequence it is called a "Series" (it sounds like another name for sequence, but it is actually a sum). See <a href="infinite-series.html">Infinite Series</a>.</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Odd numbers</h3>
|
||
<p class="center">Sequence: <b>{1, 3, 5, 7, ...}</b></p>
|
||
<p class="center">Series: <b>1 + 3 + 5 + 7 + ...</b></p>
|
||
<p class="center">Partial Sum of first 3 terms: <b>1 + 3 + 5</b></p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">595, 1242, 596, 1243, 597, 3009, 3010, 1244, 3011, 3012</div>
|
||
|
||
<div class="related">
|
||
<a href="sequences-finding-rule.html">Sequences - Finding a Rule</a>
|
||
<a href="../numberpatterns.html">Common Number Patterns</a>
|
||
<a href="sequence-diff-tool.html">Sequence Differences Tool</a>
|
||
<a href="infinite-series.html">Infinite Series</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
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