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class="page_title"> 1.2 Arithmetic Sequences and Series <a href="#" class="mark-page-favorite pull-right" data-pid="2262" 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">1.2 Arithmetic 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/AS-1.jpg" style="width: 100px; height: 100px; float: left;">1, 5, 9, 13</h2> <p>These are sequences where the difference from one term to the next remains constant, like the one in the title! Arithmetic sequences crop up in all sorts of places and this unit looks at understanding how to generalise, sum and solve problems involving arithmetic sequences.</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-f2694dd8-7fff-abc6-196c-421defd33287"></span></p> <p dir="ltr"><span id="docs-internal-guid-f2694dd8-7fff-abc6-196c-421defd33287">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-f2694dd8-7fff-abc6-196c-421defd33287">recognise arithmetic sequences</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the nth term of an arithmetic sequence</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find missing terms in an arithmetic sequence</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the sum of an arithmetic 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"> <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>Slides Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-253" class="dynamic-gallery carousel slide" data-id="253"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-253/1.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/1.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/2.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/2.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/3.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/3.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/4.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/4.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/5.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/5.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/6.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/6.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/7.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/7.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/8.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/8.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/9.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/9.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/10.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/10.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-253/11.jpg" data-fancybox="gallery-253" title="" data-caption=""><img alt="" src="../../../std-galleries/15-253/11.jpg"></a></div></div><a class="left carousel-control" href="#carousel-253" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-253" 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/15-253/1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-253/2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-253/3-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-253/4-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-253/5-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-253/6-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-253/7-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-253/8-thumb128.jpg"><li><li data-index="8"><img title="Click to view" src="../../../std-galleries/15-253/9-thumb128.jpg"><li><li data-index="9"><img title="Click to view" src="../../../std-galleries/15-253/10-thumb128.jpg"><li><li data-index="10"><img title="Click to view" src="../../../std-galleries/15-253/11-thumb128.jpg"><li></li></ol> </div> </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>1. What is an Arithmetic Sequence</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;">This video describes the nature and properties of an arithmetic sequence.<iframe allowfullscreen="" frameborder="0" height="465" mozallowfullscreen="" src="https://player.vimeo.com/video/157743921" webkitallowfullscreen="" width="640"></iframe></p> </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 an arithmetic sequence</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;">This video goes over the process of describing arithmetic sequences generally.<iframe allowfullscreen="" frameborder="0" height="460" mozallowfullscreen="" src="https://player.vimeo.com/video/157738840" webkitallowfullscreen="" width="640"></iframe></p> </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 an arithmetic sequence</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 15th 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/157737465"></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 style="text-align: center;">This video describes what sigma notation is. There are additional videos on the TI-84, Ti-Nspire and Casio pages which show you how to evaluate a series in sigma notation on the GDC. </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/665346987"></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 an arithmetic sequence</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;">How do we find the total of a given number of terms of an arithmetic sequence?<iframe allowfullscreen="" frameborder="0" height="428" mozallowfullscreen="" src="https://player.vimeo.com/video/157741468" webkitallowfullscreen="" width="640"></iframe> </p> </div> </div> <div class="panel-footer"> <div> </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-254" class="dynamic-gallery carousel slide" data-id="254"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-254/nasequences-nologos001.jpeg" data-fancybox="gallery-254" title="" data-caption=""><img alt="" src="../../../std-galleries/15-254/nasequences-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-254/nasequences-nologos002.jpeg" data-fancybox="gallery-254" title="" data-caption=""><img alt="" src="../../../std-galleries/15-254/nasequences-nologos002.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-254" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-254" 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/15-254/nasequences-nologos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-254/nasequences-nologos002-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="15-314-2262"><p></p><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 arithmetic</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 7, 11, 15, 19.......</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 10, 15.....</label></p><p><label class="checkbox"><input type="checkbox"> 1, 1, 2, 3, 5......</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 3.5, 3.75, 4, 4.25......</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 175, 350, 525, 700.......</label></p><p><label class="checkbox"><input type="checkbox"> 1, 2, 4, 8.....</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 65, 57.5, 50, 42.5, 35.......</label></p><p><label class="checkbox"><input type="checkbox"> 3, 9, 15, 20.........</label></p></div><div class="q-explanation"><p>Arithmetic sequences must have a constant difference between the terms. Four of the answers do not. The others could be arithmetic sequences because they do.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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 difference in this sequence? 2, 1.8, 1.6, 1.4 (no spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-0.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Sequence goes down by 0.2 each time so the difference is -0.2</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>The general term of an arithmetic sequence is given by U<sub>n</sub> = 3 + 4(n - 1), what is the 10th term of this sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="39"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 9 into the nth term and get U<sub>10</sub> = 3 + 4 X 9, so U<sub>10</sub> = 39</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>Consider the following sequence, Un = -2 + 0.5(n - 1), work out the 8th and 15th terms and add them together.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="6.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 8 and 15 in to the general term. The 8th term is 1.5, 15th term is 5, together they sum to 6.5</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>An arithmetic sequence begins 5, 14, 23, 32..... Which of the following is the nth term of that sequence</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 5 + 8(n - 1)</label></p><p><label class="checkbox"><input type="checkbox"> 9 + 5(n - 1)</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5 + 9(n - 1)</label></p><p><label class="checkbox"><input type="checkbox"> 9n - 5</label></p></div><div class="q-explanation"><p>Using the formula for the nth term, U<sub>n</sub> = U<sub>1</sub> + d(n - 1), substitue U<sub>1 </sub>= 5 and d = 9</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>Consider the arithmetic sequence that begins 15, 31, 47.... Calculate the sum of the the first ten terms of this sequence.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="870"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitue U<sub>1</sub> = 15 and d = 16 in to the sum formula, or work out that U<sub>10</sub> = 159 and use the other formula</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>Calculate the sum of the first 20 terms of this sequence Un = 10 - 4(n - 1)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-560"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitue U<sub>1 </sub>= 10 and d = -4 in to the sum formula, or work out that U<sub>20</sub> = -66 and use the other formula</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>An arithmetic sequence has U<sub>4</sub> = 18 and U<sub>7</sub> = 30, what is the first term of the sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> </p></div><div class="q-explanation"><p>Work out that the common difference ias added 3 times between the 4th and 7th term and this is +12 all together so d = 4. Subtract the value of d 3 times to get back to the first term.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>An arithmetic sequence has U<sub>2</sub> = 18 and U<sub>9</sub> = 53, what is the 10th term of the sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="58"> <span class="review"></span> </p></div><div class="q-explanation"><p>Work out that the common difference ias added 7 times between the 2nd and 9th term and this is +35 all together so d = 5. Add the value of d once to get from the 9th to 10th terms.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>Which is greater, </p><p>a) The sum of the sequence U<sub>n</sub> = 150 + 25(n - 1) to the first 20 terms or</p><p>b) The sum of the sequence U<sub>n</sub> = 150 + 20(n - 1) to the first 25 terms </p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> a) </label></p><p><label class="checkbox"><input class="c" type="checkbox"> b)</label></p></div><div class="q-explanation"><p>The first sum is 7750 and the second is 9750</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>Exam Style Questions</p> </div> </div> <div class="panel-body"> <div> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and 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 style="text-align: center;"><img alt="" src="../../files/NA/AS/NAP1AS1b.jpg" style="width: 600px; height: 506px;"></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"> <p style="text-align: center;"><img alt="" src="../../files/NA/AS/NAP1AS1bsol.jpg" style="width: 600px; height: 377px;"></p> </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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;"><img alt="" src="../../files/NA/AS/NAP1AS2b.jpg" style="width: 600px; height: 711px;"></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"> <p style="text-align: center;"><img alt="" src="../../files/NA/AS/NAP1AS2bsol.jpg" style="width: 600px; height: 508px;"><iframe allowfullscreen="" frameborder="0" height="375" mozallowfullscreen="" src="https://player.vimeo.com/video/141532354" webkitallowfullscreen="" width="500"></iframe> </p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-colored-body"> <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> <p><img alt="" src="../../images/dans-images/series.png" style="width: 650px; height: 360px;"></p> <p>Video solution</p> 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