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class="page_title"> 1.3 Geometric Sequences and Series <a href="#" class="mark-page-favorite pull-right" data-pid="2203" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1057/number-and-algebra.html">Number and Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">1.3 Geometric Sequences and Series</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"> <h2><img alt="" src="../../files/NA/GS/GS-1.jpg" style="width: 100px; height: 100px; float: left;">1, 3, 9, 27</h2> <p>These are sequences where you go from term to term by multiplying by a common ratio. The one in the title has a common ratio of 3 because each term is 3 times bigger than the previous one. Geometric sequences grow exponentially and should be considered with the study of exponential functions and compound interest.</p> <hr> <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"> <p><span id="docs-internal-guid-80792658-7fff-4d01-c2fc-0ca161890a6d"></span></p> <p dir="ltr"><span id="docs-internal-guid-80792658-7fff-4d01-c2fc-0ca161890a6d">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation"><span id="docs-internal-guid-80792658-7fff-4d01-c2fc-0ca161890a6d">recognise geometric sequences</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the nth term of a geometric sequence</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find missing terms in a geometric sequence</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the sum of a geometric sequence</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <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"> <h4>Slides Gallery</h4> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-240" class="dynamic-gallery carousel slide" data-id="240"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos007.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos007.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos008.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos008.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos009.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos009.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos001.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos002.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos003.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos004.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos005.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos005.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-240/nagsequences-nologos006.jpeg" data-fancybox="gallery-240" title="" data-caption=""><img alt="" src="../../../std-galleries/15-240/nagsequences-nologos006.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-240" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-240" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos007-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos008-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos009-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos001-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos002-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos003-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos004-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos005-thumb128.jpg"><li><li data-index="8"><img title="Click to view" src="../../../std-galleries/15-240/nagsequences-nologos006-thumb128.jpg"><li></li></ol> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. What is a geometric sequence?</p> </div> </div> <div class="panel-body"> <div> <p>This video describes the nature and properties of geometric sequences</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/591030963"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Finding the nth term of a geometric sequence</p> </div> </div> <div class="panel-body"> <div> <p>This video goes over the process of describing geometric sequences generally.</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/591031941"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Finding terms of geometric sequences</p> </div> </div> <div class="panel-body"> <div> <p>How to find terms with a given position in the sequence. For example, for a given sequence, how do you find the 20th term?</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/591032522"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Sigma Notation</p> </div> </div> <div class="panel-body"> <div> <p>What is sigma notation? (There is a video on sigma notation in the arithmetic sequences and series page, alongside videos on the TI-84, TI-Nspire and Casio pages of the site in terms of using yor GDC to evaluate a series in sigma notation.</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/665351263"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Finding the sum of a geometric sequence</p> </div> </div> <div class="panel-body"> <div> <p>How do find the total of a given number of terms of an arithmetic sequence?</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/591036996"></iframe></div> </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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <h4>Flash Cards</h4> <p>Review these condensed 'key point' flashcards to help you check and keep ideas fresh in your mind.</p> <div id="carousel-241" class="dynamic-gallery carousel slide" data-id="241"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-241/nagsequences-nologos001.jpeg" data-fancybox="gallery-241" title="" data-caption=""><img alt="" src="../../../std-galleries/15-241/nagsequences-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-241/nagsequences-nologos002.jpeg" data-fancybox="gallery-241" title="" data-caption=""><img alt="" src="../../../std-galleries/15-241/nagsequences-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-241/nagsequences-nologos003.jpeg" data-fancybox="gallery-241" title="" data-caption=""><img alt="" src="../../../std-galleries/15-241/nagsequences-nologos003.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-241" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-241" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-241/nagsequences-nologos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-241/nagsequences-nologos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-241/nagsequences-nologos003-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-315-2203"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sequences could be geometric?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 2, 4, 8, 16, 31</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 9, 12</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5, 2.5, 1.25, 0.625</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2, -6, 18, -36</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 9, 15</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5, 10, 20, 40</label></p></div><div class="q-explanation"><p>Onnly the sequences where each term is found by multiplying the previous term by the same common ratio. Example, 5, 10, 20 works because each term is the previous term multiplied by 2.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the common ratio of this geometric sequence?</p><p>7, 21, 63, 189</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p>7 x 3 = 21, 21 x 3 = 63, 63 x 3 = 189</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the next term of this geometric sequence?</p><p>4, 6, 9.....</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="13.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Each term is 1.5 times bigger than the previous term</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the 6<sup>th</sup> term of the geometric sequencethat begins 3, 15, 75 ?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="9375"> <span class="review"></span> </p></div><div class="q-explanation"><p>3 x 5<sup>5</sup> = 9375</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the 10th term of the geometric sequence Un = 2.5 x (-2)<sup>(n-1)</sup><span id="cke_bm_1086C" style="display: none;"> </span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-1280"> <span class="review"></span> </p></div><div class="q-explanation"><p>2.5 x (-2)<sup>9</sup></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following could be the nth term of the geometric sequence that begins...... 2, 8, 32, 128?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 4 x 2<sup>(n-1) </sup></label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2 x 4<sup>(n-1)</sup> </label></p><p><label class="checkbox"><input type="checkbox"> 4<sup>(n-1)</sup></label></p><p><label class="checkbox"><input type="checkbox"> 2 x 4<sup>n</sup></label></p></div><div class="q-explanation"><p>The general expression for the nth term of a geometric sequence is U<sub>n</sub> = U<sub>1</sub> x r<sup>(n-1)</sup>, where U<sub>1</sub> is the first term, r is the common ratio and n is the postion/number of terms. We can see that the common ratio is 4 and the first term is 2.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sequences could be a part of the geometric sequence with the general expression Un = 3 x 2<sup>(n-1)</sup>?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 6, 12, 24</label></p><p><label class="checkbox"><input type="checkbox"> 1.5, 3, 6</label></p><p><label class="checkbox"><input type="checkbox"> 30, 60, 120</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 192, 384, 768</label></p></div><div class="q-explanation"><p>6, 12, 24 are the first 3 terms. If you divide the terms 192, 384, 768 by 3 (the first term) you get 64, 128, 256, which are successive powers of 2 and so each of these terms are 3 x a power of 2 and so they must be part of the sequence.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the geometric sequence that begins 4.5, 9, 18....</p><p>What is the sum of the first 8 terms of this sequence?<span id="cke_bm_1086C" style="display: none;"> </span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1147.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 8, U1 = 4.5 and r = 2 in to the sum formula.</p><p><span class="math-tex">\({ S }_{ n }={ u }_{ 1 }\left( \frac { { r }^{ n }-1 }{ r-1 } \right) \\ in\quad this\quad case,\quad { u }_{ 1 }=4.5\quad and\quad r=2\\ SO,\\ { S }_{ 8 }=4.5\left( \frac { { 2 }^{ 8 }-1 }{ 2-1 } \right) \\ { S }_{ 8 }=4.5\left( \frac { 255 }{ 1 } \right) \\ { S }_{ 8 }=1147.5\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the sum of the first 20 terms of this geometric sequence, U<sub>n</sub> = 15 x 1.2<sup>(n - 1)</sup>. Give you answer correct to 3 sf.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="2800"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 20, U<sub>1</sub> = 15 and r = 1.2 in to the sum formula. Then round carefully.</p><p><span class="math-tex">\({ S }_{ n }={ u }_{ 1 }\left( \frac { { r }^{ n }-1 }{ r-1 } \right) \\ in\quad this\quad case,\quad { u }_{ 1 }=15\quad and\quad r=1.2\\ SO,\\ { S }_{ 20 }=15\left( \frac { { 1.2 }^{ 20 }-1 }{ 1.2-1 } \right) \\ { S }_{ 20 }=2800.319994\\ { S }_{ 20 }=2800\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>A geometric sequence has U3 = 9 and U6 = -243, what is its common ratio?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\({ u }_{ 3 }\times { r }\times { r }\times { r }={ u }_{ 6 }\\ { u }_{ 3 }\times { r }^{ 3 }={ u }_{ 6 }\\ 9\times { r }^{ 3 }=-243\\ { r }^{ 3 }=\frac { -243 }{ 9 } =-27\\ r=\sqrt [ 3 ]{ -27 } =-3\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel-footer"> <div> </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><span style="color: inherit;">Exam Style Questions</span></p> </div> </div> <div class="panel-body"> <div> <p>The following questions are based on IB exam style questions from past exams. Try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <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>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../files/NA/GS/Q1.jpg" style="width: 700px; height: 172px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <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/591044202"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" 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