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class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Geometric Sequences</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/sequences-and-series/main-1.jpg" style="float: left; width: 100px; height: 100px;"><img class="sibico" src="../../../img/sibico/hl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL"> <img class="sibico" src="../../../img/sibico/sl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL"> These are sequences where you go from term to term by <strong>multiplying </strong>by a common <strong>ratio</strong>. The sequence in the image 1, 2 , 4 , 8 has a common ratio of 2 since we multiply the previous term by 2. There are formula in the booklet to help you with this topic, but it is better not to be too reliant on them. This topic is linked to exponential growth and compound interest. Exam questions often use other areas of the mathematics course. We will get practice in all of these ideas on this page</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Use the nth term of a geometric sequence</li> <li>Use the nth term of a geometric series</li> <li>Use <span class="math-tex">\(\sum\)</span> notation for sums</li> <li>Understand applications such as compound interest and depreciation</li> <li>Find the sum of an infinite convergent series</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Understanding Geometric Sequences</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="417"> <p>Due to the style of the IB questions, it is not good to become reliant on the formula in the booklet. In the following video, we will gain a conceptual understanding of where the formula for the nth term of geometric sequences come from.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/581542515"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/geometric_sequences-conceptual.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/geometric_sequences-conceptual.pdf" width="640"></iframe></p> </section> </div> <div class="smart-object center" data-id="416"> <p>In the following applet, you can explore the effect changing <strong><em>r </em></strong>(and U<sub>1</sub>) has on the sequence.</p> <p>What happens when</p> <ol> <li>r > 1</li> <li>r = 1</li> <li>0 < r < 1</li> <li>r = 0</li> <li>-1 < r < 0</li> <li>r = -1</li> <li>r < -1</li> </ol> <p style="text-align: center"><iframe height="590px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/hrd2pttu/width/662/height/590/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Geometric sequences" width="662px"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Finding Missing Terms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="418"> <p>Being reliant on the formula for the nth term of a sequence can sometimes make the problems more difficult to solve. In the following video, it is far easier just to think about the problem conceptually.</p> <p><strong><em>A geometric sequence has second term 36 and fourth term 16.</em></strong></p> <p><strong><em>Find the first term.</em></strong></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/279258882"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/finding-missing-terms.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/finding-missing-terms.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>How many Terms?</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="419"> <p>In the following video, we look at an example in which we have to find the number of terms in a geometric sequence. This requires us to solve an equation with indices. There are several ways of doing this: using logarithms, the table function on our calculator or a graphical method. The example is about compound interest.</p> <p><em><strong>$10 000 is invested in a bank receiving 4% interest at the end of each year.</strong></em></p> <p><em><strong>How long does it take before the investment doubles in value?</strong></em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/279271949"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/how-many-terms.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/how-many-terms.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Algebraic Problem</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="420"> <p>In the following video, we need to use our algebraic skills to find an unknown</p> <p><strong><span class="math-tex">\(2k+5 \ ,\ k+5 \ ,\ k-1\)</span> <em>are consecutive terms of a geometric sequence. Find k</em></strong></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/279277023"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/geometric---problem-solving.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/geometric---problem-solving.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sum of Geometric Series</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="421"> <p>When we add terms of a sequence together, we call it a series</p> <p>geometric sequence 1 , 2 , 4 , 8</p> <p>geometric series 1 + 2 + 4 + 8</p> <p>There are two formulae for the <strong>sum to n terms</strong> of a geometric series</p> <p><span class="math-tex">\({ S }_{ n }=\frac { { U }_{ 1 }({ r }^{ n }-1) }{ r-1 } \)</span> ... useful when r > 1</p> <p><span class="math-tex">\({ S }_{ n }=\frac { { U }_{ 1 }({ 1-r }^{ n }) }{ 1-r } \)</span> ... useful when r < 1</p> <h3>Sum to Infinity</h3> <p>If -1 < r < 1 , then the series is convergent</p> <p><a href="../../files/algebra/sequences-and-series/convergent-series.jpg"><img class="pop" src="../../files/algebra/sequences-and-series/convergent-series.jpg" style="max-width: 100%; height: 174px; width: 300px;"></a></p> <p>Here, r = 0.5 and the series converges.</p> <hr> <p>When the series converges, we can find the sum to infinity</p> <p><span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ r-1 } \)</span></p> <hr class="hidden-separator"> <p>In the following video, we look at a typical exam-style question which involves the sum of a geometric series. In particular, we look at a very useful technique for solving simultaneous equations:</p> <p><em><strong>The sum of the first three terms of a geometric series is 61.</strong></em></p> <p><em><strong>The sum to infinity is 125.</strong></em></p> <p><em><strong>Find the common ratio.</strong></em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/279397193"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/sum-of-geometric-series.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/sum-of-geometric-series.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;"><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/revision-notes_geometric_sequences.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/sequences-and-series/revision-notes_geometric_sequences.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#e4372b2b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="e4372b2b"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Geometric Sequence <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-195-661" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p class="gap-drop">The sequence below are either are arithmetic, geometric or neither. Drag the correct names into the spaces</p><p class="gap-drop"><span class="q-text-draggable draggable" draggable="true">geometric</span> <span class="q-text-draggable draggable" draggable="true">arithmetic</span> <span class="q-text-draggable draggable" draggable="true">neither</span> <span class="q-text-draggable draggable" draggable="true">geometric</span> <span class="q-text-draggable draggable" draggable="true">arithmetic</span> <span class="q-text-draggable draggable" draggable="true">neither</span> <span class="q-text-draggable draggable" draggable="true">geometric</span> <span class="q-text-draggable draggable" draggable="true">arithmetic</span> <span class="q-text-draggable draggable" draggable="true">neither</span> <span class="q-text-draggable draggable" draggable="true">geometric</span> <span class="q-text-draggable draggable" draggable="true">arithmetic</span> <span class="q-text-draggable draggable" draggable="true">neither</span> <span class="q-text-draggable draggable" draggable="true">geometric</span> </p></div><div class="q-answer"><p>1 , 4 , 7 , 10 , ... <input type="text" style="height: auto;" data-c="arithmetic"> <span class="review"></span></p><p>1 , 2 , 4 , 8 , ... <input type="text" style="height: auto;" data-c="geometric"> <span class="review"></span></p><p>0.5 , 1.25 , 2 , 2.75 , ... <input type="text" style="height: auto;" data-c="arithmetic"> <span class="review"></span></p><p><span class="math-tex">\(16 \ , \ 12 \ , \ 9 \ , \ \frac{27}{4} \ ,\ ...\)</span> <input type="text" style="height: auto;" data-c="geometric"> <span class="review"></span> </p><p>1 , 1 , 2 , 3 , ... <input type="text" style="height: auto;" data-c="neither"> <span class="review"></span></p><p>2 , -6 , 18 , -54 , ... <input type="text" style="height: auto;" data-c="geometric"> <span class="review"></span></p><p>1 , -1 , 1 , -1 ... <input type="text" style="height: auto;" data-c="geometric"> <span class="review"></span></p><p>1 , 4 , 9 , 16 , ... <input type="text" style="height: auto;" data-c="neither"> <span class="review"></span></p><p>5 , -5 , -15 , -25 , ... <input type="text" style="height: auto;" data-c="arithmetic"> <span class="review"></span></p><p>-1 , 0.1 , -0.01 , 0.001 , ... <input type="text" style="height: auto;" data-c="geometric"> <span class="review"></span></p></div><div class="q-explanation"><p>1 , 4 , 7 , 10 , ... common difference, d = 3</p><p>1 , 2 , 4 , 8 , ... common ratio , r = 2</p><p>0.5 , 1.25 , 2 , 2.75 , ... common difference, d = 0.75</p><p><span class="math-tex">\(16 \ , \ 12 \ , \ 9 \ , \ \frac{27}{4} \ ,\ ...\)</span> common ratio , r = <span class="math-tex">\(\frac{12}{16}=\frac{9}{12}= \frac{\frac{27}{4}}{9}=\frac{3}{4}\)</span></p><p>1 , 1 , 2 , 3 , ... There is no common difference, there is no common ratio. It is the Fibonacci sequence.</p><p>2 , -6 , 18 , -54 , ... common ratio , r = -3</p><p>1 , -1 , 1 , -1 ... common ratio , r = -1</p><p>1 , 4 , 9 , 16 , ... There is no common difference, there is no common ratio. It is the sequence of square numbers.</p><p>5 , -5 , -15 , -25 , ... common difference, d = -10</p><p>-1 , 0.1 , -0.01 , 0.001 , ... common ratio , r = -0.1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the common ratio in the following geometric sequence</p><p>25 , 15 , 9 , ...</p><p>Write your answer as a fraction in it simplest form: <span class="math-tex">\(\frac {a}{b}\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>25 , 15 , 9 , ...</p><p><span class="math-tex">\(r = \frac {15}{25}=\frac {9}{15}=\frac {3}{5}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the common ratio in the following geometric sequence</p><p>6 , -3 , 1.5 , ...</p><p>Write your answer as a decimal</p></div><div class="q-answer"><p>r = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>6 , -3 , 1.5 , ...</p><p><span class="math-tex">\(r = \frac {-3}{6}=\frac {1.5}{-3}=-0.5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following 5 graphs are visual representations of different geometric sequences. What is the value of <strong><em>r</em></strong> , the common ratio for each one?</p><p> <span class="q-text-draggable draggable" draggable="true">r = 0</span> <span class="q-text-draggable draggable" draggable="true">r = 1</span> <span class="q-text-draggable draggable" draggable="true">r = 0.5</span> <span class="q-text-draggable draggable" draggable="true">r = 2</span> <span class="q-text-draggable draggable" draggable="true">r = -2</span> <span class="q-text-draggable draggable" draggable="true">r = -0.5</span> </p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p><img alt="" src="../../files/algebra/sequences-and-series/quiz/visualising-sequences1.png" style="width: 200px; height: 224px;"></p><p> <input type="text" style="height: auto;" data-c="r = 2"> <span class="review"></span></p></td><td><p><img alt="" src="../../files/algebra/sequences-and-series/quiz/visualising-sequences5.png" style="width: 200px; height: 224px;"></p><p> <input type="text" style="height: auto;" data-c="r = 1"> <span class="review"></span></p></td><td><p><img alt="" src="../../files/algebra/sequences-and-series/quiz/visualising-sequences2.png" style="width: 200px; height: 224px;"></p><p> <input type="text" style="height: auto;" data-c="r = -2"> <span class="review"></span></p></td></tr><tr><td><p><img alt="" src="../../files/algebra/sequences-and-series/quiz/visualising-sequences4.png" style="width: 200px; height: 224px;"></p><p> <input type="text" style="height: auto;" data-c="r = 0.5"> <span class="review"></span></p></td><td><p><img alt="" src="../../files/algebra/sequences-and-series/quiz/visualising-sequences3.png" style="width: 200px; height: 224px;"></p><p> <input type="text" style="height: auto;" data-c="r = -0.5"> <span class="review"></span></p></td><td> </td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following is a geometric sequence. Fill in the gaps</p></div><div class="q-answer"><p>5 , <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> , 40</p></div><div class="q-explanation"><p>Common ratio = 2</p><p>5 , 10 , 20 , 40</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following is a geometric sequence. Fill in the gaps</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="18"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="-12"> <span class="review"></span> , 8 , <span class="math-tex">\(-\frac {16}{3}\)</span></p></div><div class="q-explanation"><p>Common ratio = <span class="math-tex">\(-\frac {2}{3}\)</span></p><p>18 , -12 , 8 , <span class="math-tex">\(-\frac {16}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The first two terms of a geometric sequence are 4 , -8</p><p> <strong><em><span class="math-tex">\(U_{10}= a \times (-2)^{b}\)</span></em></strong></p><p>Find <strong><em>a </em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em><span class="math-tex">\(U_{n}= U_{1} \times (r)^{n-1}\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>U<sub>2</sub> and U<sub>5</sub> are terms of a geometric sequence.</p><p>U<sub>2</sub> = 9</p><p>U<sub>5</sub> = -243</p><p>Find the terms below</p></div><div class="q-answer"><p>U<sub>1</sub> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p>U<sub>8</sub> = <input type="text" style="height: auto;" data-c="6561"> <span class="review"></span></p></div><div class="q-explanation"><p>_ , 9 , _ , _ , -243</p><p><span class="math-tex">\(9 \times r^3 = -243 \\ \quad \ r^3 = \frac {-243}{9} \\ \quad \ r^3 = -27\\ \quad \ r = \sqrt [ 3 ]{ -27 } \\ \quad \ r = -3 \)</span></p><p>-3 , 9 , -27 , 81 , -243 , 729 , -2187 , <strong>6561</strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the sum of the first 6 terms of the following sequence</p><p>1.5 , 3 , 6 , ...</p></div><div class="q-answer"><p>sum = <input type="text" style="height: auto;" data-c="94.5"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a geommetric sequence with</p><p>U<sub>1</sub> = 1.5</p><p>r = 2</p><p>n = 6</p><p><span class="math-tex">\({ S }_{ n }=\frac { { U }_{ 1 }\left( { r }^{ n }-1 \right) }{ r-1 } \\ { S }_{ 6 }=\frac { 1.5\left( { 2 }^{ 6 }-1 \right) }{ 2-1 } \\ { S }_{ 6 }=\frac { 1.5\times 63 }{ 1 } \\ { S }_{ 6 }= 94.5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The first term of a geometric series is 10.</p><p>The sum to infinity is 50</p><p>Find U<sub>2</sub></p></div><div class="q-answer"><p>U<sub>2</sub> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p><span></span><span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ 1-r } \\ 50 = \frac { 10 }{ 1-r } \\ 1-r = \frac { 10 }{ 50 } \\ 1-r = \frac { 1 }{ 5 } \\ r = \frac { 4 }{ 5 } \\\)</span><span></span></p><p><span class="math-tex">\(10\div \frac { 4 }{ 5 } =8\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p> Here is a quiz about the financial applications of geometric sequences</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#8fcd7ce2"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="8fcd7ce2"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Financial Applications of geometric sequences <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-633-661" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>$5 000 is invested in a fixed deposit account with annual interest rate of 3% <strong>compounded annually</strong>. Calculate the value of the investment after 8 years.</p><p>Give your answer to the nearest dollar</p></div><div class="q-answer"><p>Value of investment = $<input type="text" style="height: auto;" data-c="6334"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large5000 \times (1+\frac{3}{100})^8\\\large =6\ 333.85\\\large =$6\ 334\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>$7 500 is invested in a fixed deposit account with annual interest rate of 1.6% <strong>compounded quarterly</strong>. Calculate the value of the investment after 5 years.</p><p>Give your answer to the nearest dollar</p></div><div class="q-answer"><p>Value of investment = $<input type="text" style="height: auto;" data-c="8123"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large7500 \times (1+\frac{1.6}{4\times100})^{5\times4}\\\large =8\ 123.36\\\large =$8\ 123\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>€ 2 500 is invested in a fixed deposit account with annual interest rate of 2.4% <strong>compounded monthly</strong>. Calculate the value of the investment after 4 years.</p><p>Give your answer to the nearest euro</p></div><div class="q-answer"><p>Value of investment = € <input type="text" style="height: auto;" data-c="2752"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large2500 \times (1+\frac{2.4}{12\times100})^{4\times12}\\\large =2\ 751.63\\\large =€2\ 752\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>€ 800 is invested in a fixed deposit account with annual interest rate of 5.8% compounded <strong>half-yearly</strong>. Calculate the value of the <strong>interest</strong> received after 10 years.</p><p>Give your answer to the nearest euro</p></div><div class="q-answer"><p>Value of interest = € <input type="text" style="height: auto;" data-c="617"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large800 \times (1+\frac{5.8}{2\times100})^{10\times2}-800\\\large =617.09\\\large =€617\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A Tesla 3 is purchased in September 2021 for $43 990. If it depreciates in value by 12% per year, calculate the value of the car in September 2025.</p><p>Give your answer to the nearest dollar</p><p><img alt="" src="../../files/algebra/sequences-and-series/financial-applications/tesla.png" style="width: 400px; height: 226px;"></p></div><div class="q-answer"><p>Value of car = $<input type="text" style="height: auto;" data-c="26381"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large43\ 990 \times (1-\frac{12}{100})^4\\\large =26\ 380.60\\\large =$26\ 381\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A Mercedes S Class is purchased for €100 600. How many years will it take for the value of the car to fall below €50 000 if it depreciates in value by 9% per year.</p><p><img alt="" src="../../files/algebra/sequences-and-series/financial-applications/mercedes.png" style="width: 400px; height: 245px;"></p></div><div class="q-answer"><p>After <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> years</p></div><div class="q-explanation"><p>After <strong><em>n</em></strong> years, value = <span class="math-tex">\(\large100 \ 600 \times (1- \frac{9}{100})^n\)</span></p><p>After 7 years, value = 51 986</p><p>After 8 years, value = 47 307</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Amir invests some money in an account that pays an annual interest rate of 4% that is compounded <strong>half-yearly</strong>.</p><p>The amount of money in Amir's account at the <strong>end of the year </strong>follows a geometric sequence with a commmon ratio <strong><em>r</em></strong>.</p><p>Find the value of <strong><em>r </em></strong>(give the exact value)</p></div><div class="q-answer"><p><em><strong>r </strong></em>= <input type="text" style="height: auto;" data-c="1.0404"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large (1+ \frac{4}{2\times100})^2=1.0404\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>On 1st January 2022, Aaliyah invests some money in an account that pays an annual interest rate of 4% that is compounded <strong>quarterly</strong>.</p>Find the year in which the amount of money in the account has doubled in value.<p> </p></div><div class="q-answer"><p>Year =<input type="text" style="height: auto;" data-c="2040"> <span class="review"></span></p></div><div class="q-explanation"><p>If $<em><strong>P</strong></em> is invested, the value of the investment after <strong><em>n</em></strong> years</p><p><span class="math-tex">\(\large P \times (1+\frac{4}{4\times100})^{4n}\\\large =P\times 1.01^{4n} \)</span></p><p>We want to find the number of years before the investment has a value of $<em><strong>2P</strong></em></p><p>We need to solve</p><p><span class="math-tex">\(\large 2P=P\times 1.01^{4n} \\\large2=1.01^{4n}\)</span></p><p>We can solve by trial and error or using logarithms</p><p><span class="math-tex">\(\large \log 2=\log{1.01}^{4n}\\ \large \log 2=4n\log{1.01}\\ \large \frac{\log2}{\log1.01}=4n\\ \large4n=69.7\\ \large n=17.4\)</span></p><p>Hence, it will take 18 years.</p><p>2022 + 18 = 2040</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>$<strong><em>P</em></strong> is invested in a fixed deposit account with annual interest rate of 5% compounded<strong> quarterly</strong>. The value of the investments after 8 years is $5208 . What was the initial amount invested?</p><p>Give your answer to the nearest dollar</p></div><div class="q-answer"><p>Initial investment = $<input type="text" style="height: auto;" data-c="3500"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large5\ 208=P \times (1+\frac{5}{4\times100})^{8\times 4}\\ \large5\ 208=P \times 1.0125^{32}\\ \large \frac{5 \ 208}{1.0125^{32}}=P\\ \large P =3\ 499.69\\\large P=$3\ 500\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A sum of money is invested in a fixed deposit account with annual interest rate of <em><strong>r</strong></em>% compounded <strong>monthly</strong>. If the investment doubles in value after 10 years, find the rate of interest, <strong><em>r</em></strong></p><p>Give your answer to the nearest whole number</p></div><div class="q-answer"><p><strong><em>r</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span> %</p></div><div class="q-explanation"><p>If <strong><em>P </em></strong> is invested, we are trying to find the rate of interest that will make the value be <strong><em>2P</em></strong> after 10 years.</p><p>We need to solve</p><p><span class="math-tex">\(\large2P=P \times (1+\frac{r}{12\times100})^{10\times12}\\\)</span></p><p>We can see that the value of the initial investment is irrelevant. We need to solve</p><p><span class="math-tex">\(\large2= (1+\frac{r}{12\times100})^{10\times12}\\ \large 2= (1+\frac{r}{1200})^{120}\\ \large \sqrt[120]{2}=1+\frac{r}{1200}\\ \large (\sqrt[120]{2}-1)\times1200=r\\ \large r=6.95\\ \large r=7\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="408"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The nth term of a geometric sequence is U<sub>n</sub> , where Un<sub> </sub>=<span class="math-tex">\(48(\frac{1}{4})^n\)</span></p> <p>a) Find U<sub>1</sub></p> <p>b) Find the sum to infinity of the series</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) let n = 1</p> <p>b) <span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ 1-r } \)</span><content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="409"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The first term of a geometric series is 10. The sum to infinity is 50.</p> <p>a) Find the common ratio</p> <p>The nth term is U<sub>n</sub></p> <p>b) Find the value of <span class="math-tex">\(\sum _{ n=1 }^{ 20 }{ { U }_{ n } } \)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) <span></span><span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ 1-r } \)</span><span></span></p> <p>b) You can use your graphical calculator to find the sum<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="410"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Three terms of a geometric sequence are <span class="math-tex">\(x+6 \ , \ 12 \ ,\ x-1\)</span></p> <p>Find the possible values of<em><strong> x</strong></em></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Common ratio , <span class="math-tex">\(r=\frac{U_2}{U_1}=\frac{U_3}{U_2}\)</span><content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="411"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Consider the geometric sequence where the first term is 45 and the second term is 36.</p> <p>a) Find the least value of <strong><em>n</em></strong> such that the nth term of the sequence is less than 1</p> <p>b) Find the least value of <strong><em>n</em></strong> such that the sum of the first <em><strong>n</strong></em> terms of the sequence is more than 200.</p> <p>c) Find the sum to infinity.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>You can make good use of your calculator to solve the inequalties in part a and b.</p> <p><content>The table function is particularly useful.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="412"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The sum to infinity of a geometric series is 27.</p> <p>The sum of the first 3 terms is 19.</p> <p>Find the common ratio</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p>We need to solve simultaneous equations.</p> <p>Notice that <span class="math-tex">\(\frac { { U }_{ 1 } }{ 1-r } \)</span> is common to both equations</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric5a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric5a.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="413"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The 2nd, 3rd and 6th terms of an <strong>arithmetic</strong> sequence with common difference <span class="math-tex">\(d, \ d\neq 0\)</span> form the first three terms of a <strong>geometric</strong> sequence, with common ratio, <strong><em>r</em></strong>.</p> <p>The 1st term of the arithmetic sequence is <strong><em>-2</em></strong>.</p> <p>a) Find <strong><em>d</em></strong>.</p> <p>The sum of the first <strong><em>n</em></strong> terms of the geometric sequence exceeds the sum of the first <strong><em>n</em></strong> terms of the arithmetic sequence by at least 1000.</p> <p>b) Find the least value of <strong><em>n</em></strong> for which this occurs.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Geometric sequence is U<sub>2</sub> , U<sub>3</sub> , U<sub>6</sub></p> <p><content>common ratio , <span class="math-tex">\(r=\frac{U_2}{U_1}=\frac{U_3}{U_2}\)</span></content></p> <p>b) You can make good use of the table function on your graphical calculator to solve</p> <p>(sum to n terms of geometric sequence) - (sum to n terms of arithmetic sequence) > 1000</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric6.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="414"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>U<sub>1</sub> = cosx ,U<sub>2</sub> = sin2x are the first two terms of a geometric sequence, <span class="math-tex">\(-\frac{\pi}{2}<x<\frac{\pi}{2}\)</span></p> a) Find U<sub>3</sub> in terms of cosx <p> </p> <p> </p> <p>b) Find the set of values of <strong><em>x</em></strong> for which the geometric series converges</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) <span class="math-tex">\(sin2x\equiv 2sinxcosx\)</span></p> <p>b) For a series to converge <em><strong>-1 < r < 1</strong></em></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric7.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 8</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="415"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>a) Jessica takes out a loan of $200 000 to buy an appartment. The interest rate is 4% and is calculated at the end of each year. Calculate to the nearest dollar the amount Jessica would owe the bank after 15 years.</p> <p>b) In order to pay of the loan, she pays $<strong><em>P</em></strong> into a bank at the end of <strong>each </strong>year. She receives an interest rate of 2.5% per year. Find the amount saved after 15 years.</p> <p>c) What must be the value of <strong><em>P </em></strong>so that she has saved enough money to pay off the loan.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) This is a geometric sequence</p> <p>b) This is a geometric series</p> <p>c) In order to pay of the loan</p> <p>Amount of savings = Amount of loan</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/sequences-and-series/esq_geometric8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_geometric8.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="661"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Geometric Sequences</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center 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