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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">Arithmetic 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/maina-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>adding </strong>a common <strong>difference</strong>. The sequence in the image 1, 5 , 9 has a common difference of 4 since we add 4 to the previous term. There are formula in the booklet to help you with this topic, but it is better not to be too reliant on them. 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 formula for the nth term of an arithmetic sequence</li> <li>Use the formula for the nth term of an arithmetic series</li> <li>Use <span class="math-tex">\(\sum\)</span> notation for sums</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>Arithmetic Sequence - nth Term</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="424"> <p>In the following video, we will see where the formula for the nth term of an arithmetic sequence comes 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/279618211"></iframe></div> <h4><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>Print from <a href="../../files/algebra/sequences-and-series/arithmetic-sequences---nth-term.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/arithmetic-sequences---nth-term.pdf" width="640"></iframe></p> </section> </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>Arithmetic Sequences - Finding Terms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="423"> <p>In the following video, we look at a typical exam-style question about arithmetic sequences. We do not have to be too reliant on the formula for the nth term to be able to find terms in a sequence.</p> <p><em><strong>The 5th term of an arithmetic sequence is 35 and the 10th term is 85.</strong></em></p> <p><em><strong>What is the 20th term?</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/279612251"></iframe></div> <h4><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>Print from <a href="../../files/algebra/sequences-and-series/arithmetic-sequences---finding-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/arithmetic-sequences---finding-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-border panel-expandable panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Arithmetic Sequence - Problem Solving</p> </div> </div> <div class="panel-body"> <div> <div class="panel-body"> <div class="smart-object center" data-id="422"> <p>In the following video, we will look at a typical exam-style question. We can use the information in the question to set up and solve two equations. We are going to concentrate on <strong>how</strong> we can make the equations easier to solve.</p> <p><em><strong>Three numbers are consecutive terms in an arithmetic sequence.</strong></em></p> <p><em><strong>They add to give 45 and when they are multiplied together we get 2640.</strong></em></p> <p><em><strong>What are the three numbers?</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/279608237"></iframe></div> <h4><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>Print from <a href="../../files/algebra/sequences-and-series/arithmetic-sequences---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/arithmetic-sequences---problem-solving.pdf" width="640"></iframe></p> </section> </div> </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>Arithmetic Series</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="425"> <p>When we add terms of a sequence together, we call it a series:</p> <p>arithmetic sequence 1 , 4 , 7 , 10</p> <p>arithmetic series 1 + 4 + 7 + 10</p> <p>There are two useful formulae for the <strong>sum to n terms</strong> of a arithmetic series</p> <p><span class="math-tex">\({ S }_{ n }=\frac { n }{ 2 } \left( { U }_{ 1 }+{ U }_{ n } \right) \)</span></p> <p><span class="math-tex">\({ S }_{ n }=\frac { n }{ 2 } \left( { 2U }_{ 1 }+(n-1)d \right) \)</span></p> <hr class="hidden-separator"> <p>In the following video, we look at a typical exam-style question which involves the sum of an arithmetic series.</p> <p><em><strong>The sum of the first 11 terms of an arithmetic series is 3 times the sum of the first 5 terms.</strong></em></p> <p><em><strong>The 8th term is 53. Find the common difference.</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/279629945"></iframe></div> <h4><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>Print from <a href="../../files/algebra/sequences-and-series/arithmetic-sequences---sum.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/arithmetic-sequences---sum.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"> <div> <p>How many Terms?</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="426"> <p>In the following video, we look at a typical exam-style question which involves the sum of an arithmetic series.</p> <p>For the following sequence <span class="math-tex">\(28 , 25 , 22 ,...\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Which is the first negative term of the sequence?</li> <li>Which is the first term of the series that makes the sum of the series become negative?</li> </ol> <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/279793433"></iframe></div> <h4><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>Print from <a href="../../files/algebra/sequences-and-series/arithmetic-sequence---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/arithmetic-sequence---how-many-terms.pdf" width="640"></iframe></p> </section> </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 pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/revision-notes_arithmetic_sequences.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/sequences-and-series/revision-notes_arithmetic_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="#71f717d0"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="71f717d0"> <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%;"> Arithmetic 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="11-194-662" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The following is an arithmetic sequence. Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p><p><em><strong>a</strong></em> , 5 , _ , 11 , _ , _ , <em><strong>b</strong></em></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="20"> <span class="review"></span></p></div><div class="q-explanation"><p><em><strong>2</strong></em> , 5 , 8 , 11 , 14 , 17 , <em><strong>20</strong></em></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>In an arithmetic sequence, U<sub>3</sub> = 18 and U<sub>6</sub> = 6</p><p>Find U<sub>1</sub></p></div><div class="q-answer"><p>U<sub>1</sub> = <input type="text" style="height: auto;" data-c="26"> <span class="review"></span></p></div><div class="q-explanation"><p>26, 22 , 18 , 14 , 10</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 , 1 , 4 , ...</p><p>The nth term, U<sub>n</sub> = <strong><em>a</em></strong> +(n - 1)<strong><em>d</em></strong></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><strong><em>d </em></strong>= <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>U<sub>n</sub> = U<sub>1</sub> + (n - 1)d</p><p>U<sub>1 </sub>= first term</p><p>d = common difference</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>1 </sub>and U<sub>5 </sub>are the first and fifth terms of an arithmetic sequence</p><p>U<sub>1 </sub>=11</p><p>U<sub>5 </sub>= 39</p><p>U<sub>10</sub> = <strong><em>a </em></strong>+ 9 <strong><em>d</em></strong></p></div><div class="q-answer"><p><em><strong>a </strong></em>= <input type="text" style="height: auto;" data-c="11"> <span class="review"></span> </p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>U<sub>5 </sub>- U<sub>1</sub> = 4d</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>7</sub> are terms of an arithmetic sequence</p><p>U<sub>2</sub> = - 2</p><p>U<sub>7</sub> = 23</p></div><div class="q-answer"><p>U<sub>11</sub> = <input type="text" style="height: auto;" data-c="43"> <span class="review"></span></p></div><div class="q-explanation"><p>23 - (-2)= 5d</p><p>25 = 5d</p><p>5 = d</p><p>U<sub>11</sub> = U<sub>7 </sub>+ 4d</p><p>U<sub>11</sub> = 23<sub> </sub>+ 20</p><p>U<sub>11</sub> = 43</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>32 , 29 , 26 , ...</p><p>The nth term of this sequence is -4</p><p>Find n</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span></p></div><div class="q-explanation"><p>d = -3</p><p>-4 - 32 = -36</p><p>-36 = 12(-3)</p><p>There are 12 differences. It is the 13th term</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>An arithmetic series has a first term of 8 and a common difference of 4.</p><p>What is the sum of the first 12 terms?</p><p>S<sub>12</sub> = <strong><em>a</em></strong> (<strong><em>b</em></strong> + 11<strong><em>c</em></strong>)</p><p>Find <strong><em>a</em></strong> , <strong><em>b</em></strong> and <strong><em>c</em></strong></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span></span><span class="math-tex">\(S_{ n }=\frac { n }{ 2 } (2U_{ 1 }+(n−1)d)\)</span><span></span></p><p><span></span><span class="math-tex">\(S_{ 12}=\frac { 12 }{ 2 } (2 \times 8+(12−1)4)\)</span><span></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 and second terms of an arithmetic sequence are 4 and 9. The nth term is 79.</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p>S<sub>n</sub> = <input type="text" style="height: auto;" data-c="664"> <span class="review"></span></p></div><div class="q-explanation"><p>d = 5</p><p>79 - 4 = 75</p><p>75 = <em><strong>15</strong></em> x 5</p><p>There are <strong><em>15 </em></strong> differences. 79 is the 16th term.</p><p><span></span><span class="math-tex">\(S_{ n }=\frac { n }{ 2 } (U_{ 1 }+U_{ n})\)</span><span></span></p><p><span class="math-tex">\(S_{ 16 }=\frac { 16 }{ 2 } (4+79)= 664\)</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 sum of the following arithmetic series</p><p>(-5) + (-1) + ... + 71</p></div><div class="q-answer"><p>sum = <input type="text" style="height: auto;" data-c="660"> <span class="review"></span></p></div><div class="q-explanation"><p>d = 4</p><p>Find the number of terms in the sequence:</p><p>71 - (-5) = 76</p><p>76 = <strong>19</strong>x4</p><p>There are 19 differences, so 71 is the 20th term. n = 20</p><p><span class="math-tex">\(S_{ n }=\frac { n }{ 2 } (U_{ 1 }+U_{ n})\)</span></p><p><span class="math-tex">\(S_{ 20 }=\frac { 20 }{ 2 } (-5+71)= 660\)</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>S<sub>n</sub> is the sum of the first n terms of an arithmetic series</p><p>S<sub>1</sub> = 1 , S<sub>2</sub> = 5</p><p>Find S<sub>4</sub></p></div><div class="q-answer"><p>S<sub>4</sub> = <input type="text" style="height: auto;" data-c="22"> <span class="review"></span></p></div><div class="q-explanation"><p>S<sub>1</sub> = 1 , S<sub>2</sub> = 5</p><p>U<sub>1</sub> = 1 , U<sub>2</sub> = 4</p><p>Sequence is 1 , 4 , 7 , 10</p><p>Sum = 1 + 4 + 7 + 10 = 22</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="401"> <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>In an arithmetic sequence, the <strong>first term</strong> is 4 and the <strong>third term</strong> is 16.</p> <p>a) Find the common difference</p> <p>b) Find the 8th term</p> <p>c) Find the sum of the first 8 terms</p> <h4>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) How many differences are there between the first and the third term?</p> <p><content>b) U<sub>n </sub>= U<sub>1</sub> + (n-1)d</content></p> <p>c) <span class="math-tex">\({ S }_{ n }=\frac { n }{ 2 } \left( { U }_{ 1 }+{ U }_{ n} \right) \)</span></p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic1.pdf" width="640"></iframe></p> </section> </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 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="402"> <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 consecutive terms of an arithmetic sequence are <strong><span class="math-tex">\(x-3 \ , \ 12 \ ,\ 3x-5\)</span></strong></p> <p>Find <strong><em><span class="math-tex">\(x\)</span></em></strong></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>d = U<sub>2</sub> - U<sub>1</sub></p> <p>d = U<sub>3</sub> - U<sub>2</sub><content></content></p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic2.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="403"> <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>The 2nd term of an arithmetic sequence is 19 and the 5th term is 37.</p> <p>a) Find the 10th term</p> <p>b) The sum of the first <strong><em>n</em></strong> terms of this sequence exceeds 1000. Find the least value of <strong><em>n</em></strong></p> <h4>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) Find <em><strong>d</strong></em> by considering how many differences there are between the 2nd and 5th terms</p> <p><content>b) You can solve this algebraically or you could make good use of the table function in the your graphical calculator.</content></p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic3.pdf" width="640"></iframe></p> </section> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="404"> <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>Find the sum of all the integers between 100 and 1000 that are divisible by 9</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The challenge is to recognise that this is a question that requires finding the <strong>sum of an arithmetic</strong> series.</p> <p>What is the first integer after 100 that is divisible by 9?</p> <p>What is the last integer before 1000 that is divisible by 9?</p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic4.pdf" width="640"></iframe></p> </section> </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="405"> <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>An arithmetic sequence has first term <em><strong>U<sub>1</sub></strong></em> and common difference <strong><em>d</em></strong>. The sum of the first 17 terms is 136.</p> <p>a) Show that<strong><em> </em></strong> <span class="math-tex">\(U_1+8d=8\)</span></p> <p>The sum of the 2nd and the 3rd terms is 42.</p> <p>b) Find <strong><em>d</em></strong>.</p> <p>The nth term of the sequence is <em><strong>U<sub>n</sub></strong></em>.</p> <p>c) Find the value of <span class="math-tex">\(\sum _{ r=4 }^{ 17 }{ { U }_{ n } } \)</span></p> <h4>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">\(S_{ n }=\frac { n }{ 2 } (2U_{ 1 }+(n−1)d)\)</span></p> <p>b) <span class="math-tex">\(U_{ 2 }+U_{ 3 }\ =\ (U_{ 1 }+d)+(U_{ 1 }+2d)\)</span></p> <p>c) sum of the first 17 terms - sum of the first 3 terms</p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic5.pdf" width="640"></iframe></p> </section> </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="406"> <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>In an arithmetic sequence, the 9th term is 4 times the 5th term. The sum of the first 2 terms is -13.</p> <p>Find the 10th term</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Set up and solve two simultaneous equations using the information given:</p> <p>U<sub>9</sub> = 4U<sub>5</sub> and S<sub>2</sub> = -13</p> <p>There is no need to use the full formula for the information about the sum since S<sub>2</sub> = U<sub>1</sub> + U<sub>2</sub></p> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic6.pdf" width="640"></iframe></p> </section> </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 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="407"> <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 first terms of a sequence are log<sub>3</sub> x , log<sub>3</sub> x<sup>2</sup> , log<sub>3</sub> x<sup>3</sup> , ...</p> <p>Find <strong><em>x </em></strong>if the sum of the first 9 terms is 135</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <ul> <li>The terms in this sequence are <strong>arithmetic</strong>.</li> <li>This question requires good knowledge of the laws of logarithms</li> </ul> <p style="margin-left: 80px;">log<sub>c</sub> a + log<sub>c</sub> b = log<sub>c</sub> ab</p> <p style="margin-left: 80px;">log<sub>c</sub> a - log<sub>c</sub> b = log<sub>c</sub> <span class="math-tex">\(\frac{a}{b}\)</span></p> <ul> <li>You should be able to find an expression fo rthe common difference and then use this</li> </ul> </section> <h4>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>Print from <a href="../../files/algebra/sequences-and-series/esq_arithmetic7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/sequences-and-series/esq_arithmetic7.pdf" 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