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362 lines
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Partial Sums</h1>
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<p>A Partial Sum is a <b>Sum</b> of <b>Part</b> of a <a href="sequences-series.html">Sequence</a>.</p>
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<div class="example">
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<h3>Example: </h3>
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<p>This is the <b>Sequence</b> of even numbers from 2 onwards: <span class="large">{2, 4, 6, 8, 10, 12, ...}</span></p>
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<p>This is the <b>Partial Sum</b> of the first 4 terms of that sequence: <span class="large">2+4+6+8 = 20</span></p>
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</div>
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<p>Let us define things a little better now:</p>
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<p class="dotpoint">A <b>Sequence</b> is a set of things (usually numbers) that are in order. </p>
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<p class="center"><img src="images/sequence.svg" alt="Sequence"></p>
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<p class="dotpoint">A <b>Partial Sum</b> is the sum of <b>part</b> of the sequence.</p>
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<p> </p>
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<div class="words">
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<p>The sum of <b>infinite</b> terms is an <a href="infinite-series.html">Infinite Series</a>.</p>
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<p>And Partial Sums are sometimes called <b>"Finite Series"</b>.<br>
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</p>
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</div>
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<p> </p>
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<h2>Sigma</h2>
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<p>Partial Sums are often written using <span class="times" style="font-size:200%;">Σ</span> to mean "add them all up":</p>
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<div class="center def"><span class="times" style="font-size:400%; vertical-align:middle;">Σ</span> This symbol (called <a href="sigma-notation.html">Sigma</a>) means "sum up"</div>
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<br> <p>So <span class="times" style="font-size:200%;">Σ</span> means to sum things up ...</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td colspan="3">
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<h3>Sum What?</h3></td>
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</tr>
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<tr>
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<td>
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<p>Sum whatever is after the Sigma:</p></td>
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<td> </td>
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<td style="background-color: #def;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto"> </div>
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<div class="intsymb">Σ</div>
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<div class="intfrom"> </div>
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</div>
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<div class="inttext">n</div>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td>so we sum <span class="number"><i><b>n</b></i></span>
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</td></tr>
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<tr>
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<td colspan="3">
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<h3>But What Values of <i><b>n</b></i> ?</h3></td>
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</tr>
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<tr>
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<td>
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<p align="right">The values are shown below<br>
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and above the Sigma:</p></td>
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<td> </td>
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<td style="background-color: #def;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">n</div>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> it says <span class="number"><i>n</i></span> goes from 1 to 4,<br>
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which is <b>1</b>, <b>2</b>, <b>3</b> and <b>4</b>
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</td></tr>
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<tr>
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<td colspan="3">
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<h3>OK, Let's Go ...</h3></td>
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</tr>
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<tr>
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<td>
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<p align="right">So now we add up 1,2,3 and 4:</p></td>
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<td> </td>
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<td style="background-color: #def;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">n = 1 + 2 + 3 + 4 = <b>10</b></div>
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</td>
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</tr>
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</tbody></table><p>Here it is in one diagram:</p>
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<p align="center"><img src="images/sigma-notation.svg" alt="Sigma Notation"></p>
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<h2>More Powerful</h2>
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<p>But <span style="font-size:150%; font-family: 'Times New Roman', Times, serif; ">Σ</span> can do more powerful things than that!</p>We can square <span class="number"><i>n</i></span> each time and sum the result:
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext" style="transform: translateY(-100%);">n<sup>2</sup> = 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> = 30</div>
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</div>
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<p align="center"> </p>
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<p>We can add up the first four terms in the <a href="sequences-series.html">sequence</a> <b>2n+1</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">(2n+1) = 3 + 5 + 7 + 9 = 24</div>
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</div>
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<p> </p>
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<p>And we can use other letters, here we use <span class="larger">i</span> and sum up <span class="larger">i × (i+1)</span>, going from <b>1</b> to <b>3</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">3</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">i=1</div>
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</div>
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<div class="inttext">i(i+1) = 1×2 + 2×3 + 3×4 = 20</div>
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</div>
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<p> </p>
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<p>And we can start and end with any number. Here we go from <b>3</b> to <b>5</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">5</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">i=3</div>
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</div>
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<div class="inttext" style="transform: translateY(-70%);"><span class="intbl"><em>i</em><strong>i + 1</strong></span> = <span class="intbl"><em>3</em><strong>4</strong></span> + <span class="intbl"><em>4</em><strong>5</strong></span> + <span class="intbl"><em>5</em><strong>6</strong></span></div>
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</div>
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<h2>
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Properties
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</h2>
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<p>Partial Sums have some useful properties that can help us do the calculations.</p>
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<h3>Multiplying by a Constant Property</h3>
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<p>Say we have something we want to sum up, let's call it <span class="large">a<sub>k</sub></span></p>
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<p class="center80" align="center"><span class="large">a<sub>k</sub></span> could be <b>k<sup>2</sup></b>, or <b>k(k-7)+2</b>, or ... anything really</p>
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<p>And <b>c</b> is some constant value (like <b>2</b>, or <b>-9.1</b>, etc), then:</p>
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<p align="center"><img src="images/partial-sum-a.gif" alt="Sigma" height="56" width="161"></p>
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<p>In other words: if every term we are summing is multiplied by a constant, we can "pull" the constant outside the sigma.</p>
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<div class="example">
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<h3>Example: </h3>
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<p align="center"><img src="images/partial-sum-a1.gif" alt="Sigma" height="56" width="163"></p>
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<p>So instead of summing <b>6k<sup>2</sup></b> we can sum <b>k<sup>2</sup></b> and then multiply the whole result by <b>6</b></p>
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</div>
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<h3> </h3>
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<h3>Adding or Subtracting Property</h3>
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<p>Here is another useful fact: </p>
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<p align="center"><img src="images/partial-sum-b.gif" alt="Sigma" height="56" width="266"></p>
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<p>Which means that when two terms are added together, and we want to sum them up, we can actually sum them <b>separately</b> and then <b>add </b>the results.</p>
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<div class="example">
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<h3>Example: </h3>
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<p src="images/partial-sum-b2.gif" align="center"><img src="images/partial-sum-b2.gif" alt="Sigma" height="56" width="257"></p>
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<p>It is going to be easier to do the two sums and then add them at the end.</p>
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</div>
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<p>Note this also works for subtraction:</p>
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<p align="center"><img src="images/partial-sum-b3.gif" alt="Sigma" height="56" width="267"></p>
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<h2>Useful Shortcuts</h2>
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<p>And here are some useful shortcuts that make the sums <b>a lot easier</b>.</p>
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<p>In each case we are trying to sum from <b>1</b> to some value <b>n</b>.</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><img src="images/partial-sum-c.svg" alt="Sigma"></td>
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<td> </td>
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<td>Summing <b>1</b> <i>equals</i> <b>n</b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-d.svg" alt="Sigma"></td>
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<td> </td>
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<td>Summing the constant <b>c</b> <i>equals</i> <b>c</b> times <b>n</b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-e.svg" alt="Sigma"></td>
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<td> </td>
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<td>A shortcut when summing <b>k</b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-f.svg" alt="Sigma"></td>
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<td> </td>
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<td>A shortcut when summing <b>k<sup>2</sup></b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-g.svg" alt="Sigma"></td>
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<td> </td>
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<td>A shortcut when summing <b>k<sup>3</sup></b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-g2.svg" alt="Sigma"></td>
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<td> </td>
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<td>Also true when summing <b>k<sup>3</sup></b></td>
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</tr>
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<tr>
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<td><img src="images/partial-sum-odd.svg" alt="Sigma"></td>
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<td> </td>
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<td>Summing odd numbers</td>
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</tr>
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</tbody></table>
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<p>Let's use some of those:</p>
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<div class="example">
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<h3>Example 1: You sell concrete blocks for landscaping. </h3>
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<p>A customer says they will buy the entire "pyramid" of blocks you keep out front. The stack is 14 blocks high.</p>
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<p>How many blocks are in there?</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/partial-sum-ex1-a.svg" alt="Sigma"></p>
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<p>Each layer is a square, so the calculation is:</p>
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<p class="larger" align="center">1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + ... + 14<sup>2</sup></p>
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<p>But this can be written <b>much more easily</b> as:</p>
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<p align="center"><img src="images/partial-sum-ex1-b.gif" alt="Sigma" height="59" width="55"></p>
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<p>We can use the formula for <b>k<sup>2</sup></b> from above:</p>
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<p align="center"><img src="images/partial-sum-ex1-c.gif" alt="Sigma" height="59" width="329"></p>
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<p>That was a lot easier than adding up <b>1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + ... + 14<sup>2</sup></b>.</p>
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</div>
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<p>And here is a more complicated example:</p>
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<div class="example">
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<h3>Example 2: The customer wants a better price. </h3>
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<p>The customer says the blocks on the outside of the pyramid should be cheaper, as they need cleaning. </p>
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<p>You agree to:</p>
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<ul>
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<li>$7 for outer blocks </li>
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<li>and $11 for inner blocks.</li>
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</ul>
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<p>What is the total cost?</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/partial-sum-ex2-a.svg" alt="Sigma"></p>
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<p>You can calculate how many "inner" and "outer" blocks in any layer (except the first) using</p>
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<ul>
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<li>outer blocks = <b>4×(size-1)</b></li>
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<li>inner blocks = <b>(size-2)<sup>2</sup></b></li>
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</ul>
|
|
|
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<p>And so the cost per layer is:</p>
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<ul>
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<li>cost (outer blocks) = $7 × 4(size-1)</li>
|
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<li>cost (inner blocks) = $11 × (size-2)<sup>2</sup></li>
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</ul>
|
|
|
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<p>So all layers together (except first) will cost:</p>
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<p align="center"><img src="images/partial-sum-ex2-b.gif" alt="Sigma" height="59" width="304"></p>
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<p>Now we have the sum, let us try to make the calculations easier!</p>
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<p> </p>
|
|
<p>Using the "Addition Property" from above:</p>
|
|
<p align="center"><img src="images/partial-sum-ex2-c.gif" alt="Sigma" height="59" width="315"></p>
|
|
<p>Using the "Multiply by Constant Property" from above:</p>
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|
<p align="center"><img src="images/partial-sum-ex2-d.gif" alt="Sigma" height="55" width="312"></p>
|
|
<p>That is good ... but we can't use any shortcuts as it is, as we are going from <b>i=2</b> instead of<b> i=1</b></p>
|
|
<p>HOWEVER, if we invent two new variables:</p>
|
|
<ul>
|
|
<li>j = i-1</li>
|
|
<li>k = i-2</li>
|
|
</ul>
|
|
<p>We have:</p>
|
|
<p align="center"><img src="images/partial-sum-ex2-e.gif" alt="Sigma" height="59" width="205"></p>
|
|
<p>(I dropped the k=0 case, because I know that 0<sup>2</sup>=0)</p>
|
|
<p> </p>
|
|
<p>And now we can use the shortcuts:</p>
|
|
<p align="center"><img src="images/partial-sum-ex2-f.gif" alt="Sigma" height="49" width="337"></p>
|
|
<p>After a little calculation:</p>
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|
<p class="large" align="center">$7 × 364 + $11 × 650 = $9,698.00</p>
|
|
<p>Oh! And don't forget the top layer (size=1) which is just one block. Maybe you can give them that one for free, you are so generous!</p>
|
|
<p> </p>
|
|
<p><i>Note: as a check, when we add the "outer" and "inner" blocks, plus the one on top, we get</i></p>
|
|
<p align="center"><i><b> 364 + 650 + 1 = 1015</b></i></p>
|
|
<p><i>Which is the same number we got for the "total blocks" before ... yay!</i></p>
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</div>
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<p> </p>
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