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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">Proof by Induction</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/proof-by-induction/main_proof_image.jpg" style="float: left; width: 100px; height: 100px;">Proof by Induction is a method of proof commonly used in HL mathematics. The method is always the same and questions are worth a good deal of marks in an exam. Therefore, it is really worth investing time to understand how to use it! Questions involving series, divisibility and inequalities are usually fairly straightforward. However, it is also used to make proofs in work on calculus, trigonometry and complex numbers and these tend to be really quite challenging.</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>Carry out proof by induction for a wide variety of topics including <ul> <li>series</li> <li>divisibility</li> <li>inequalities</li> <li>differentiation</li> <li>complex numbers</li> </ul> </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>Proof by Induction for Series</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="441"> <p>Here is an example of a proof by induction that proves the formula for the sum of cubic numbers:</p> <p>Prove that <span class="math-tex">\(1^3+2^3+3^3+ ...+n^3=(\frac{n(n+1)}{2})^2 \quad, n\in \mathbb{N}\)</span></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/280155487"></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/proof-by-induction/proof-by-induction---series.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof-by-induction---series.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>Proof by Induction for Divisibility</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="438"> <p>Here is a video to show you the method to see if an expression is divisible by a certain quantity</p> <p>In the following video, we look at an example of a proof by induction for divisibility</p> <p><em><strong>Prove that <span class="math-tex">\(n^3+11n\)</span> is divisible by 3 for <span class="math-tex">\(n\in\mathbb{Z} ,n>0\)</span></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/214284076"></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/proof-by-induction/proof-by-induction-divisibility-notes.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof-by-induction-divisibility-notes.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>Proof by Induction for Inequalities</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="439"> <p>The following video shows an example of a proof by induction that includes an inequality</p> <p>Prove that <span class="math-tex">\(n!>2^n\ , \quad n\ge 4\)</span></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/193272985"></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/proof-by-induction/proof-by-induction-inequalities-notes.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof-by-induction-inequalities-notes.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> Proof by Induction for Calculus</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="440"> <p>In the following video we look at an example of proof by induction for differentiation:</p> <p>Let <span class="math-tex">\(y = \frac{1}{1-x}\)</span> , <span class="math-tex">\(x\in \mathbb{R}\)</span></p> <p>Prove by induction that <span class="math-tex">\(\frac{d^ny}{dx^n}=\frac{n!}{(1-x)^{n+1}}\)</span> , <span class="math-tex">\(n\in \mathbb{Z^+}\)</span></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/280145898"></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/proof-by-induction/proof_by_induction-differentiation.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof_by_induction-differentiation.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> Proof by Induction for Complex Numbers</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="442"> <p>In the following video, we look at an example of a proof by induction question applied to complex numbers. In it, we prove De Moivre's Theorem:</p> <p>Let <span class="math-tex">\(z=r(cosθ+isinθ)\)</span></p> <p>Prove that <span class="math-tex">\(z^{ n }≡r^{ n }[cos(nθ)+isin(nθ)]\ ,\ n\in \mathbb{Z^+}\)</span></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/280215846"></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/proof-by-induction/proof-by-induction---complex-numbers.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof-by-induction---complex-numbers.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/proof-by-induction/revision-notes_proof_by_induction.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/proof-by-induction/revision-notes_proof_by_induction.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"> <div> <p>Here is a quiz that practises the skills from this page</p> </div> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#763ca901"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="763ca901"> <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%;"> Proof by Induction <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="11-198-664" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Put the 5 steps to a proof by induction in the correct order</p><p><span class="q-text-draggable draggable" draggable="true">Let P(n) be the proposition...</span> <span class="q-text-draggable draggable" draggable="true">Show true for n=1</span> <span class="q-text-draggable draggable" draggable="true">Assume true for n=k</span> <span class="q-text-draggable draggable" draggable="true">Show true for n=k+1</span> <span class="q-text-draggable draggable" draggable="true">Concluding statement</span></p></div><div class="q-answer"><ol><li> <input type="text" style="height: auto;" data-c="Let P(n) be the proposition..."> <span class="review"></span></li><li> <input type="text" style="height: auto;" data-c="Show true for n=1"> <span class="review"></span></li><li> <input type="text" style="height: auto;" data-c="Assume true for n=k"> <span class="review"></span></li><li> <input type="text" style="height: auto;" data-c="Show true for n=k+1"> <span class="review"></span> </li><li> <input type="text" style="height: auto;" data-c="Concluding statement"> <span class="review"></span></li></ol></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>Fill in the gaps to the following proof by induction</p><p> <span class="q-text-draggable draggable" draggable="true">2k-1</span> <span class="q-text-draggable draggable" draggable="true">(k+1)²</span> <span class="q-text-draggable draggable" draggable="true">Assume</span> <span class="q-text-draggable draggable" draggable="true">Show</span> <span class="q-text-draggable draggable" draggable="true">n=k+1</span> <span class="q-text-draggable draggable" draggable="true">2k+1</span> <span class="q-text-draggable draggable" draggable="true">RHS</span> <span class="q-text-draggable draggable" draggable="true">LHS</span> </p></div><div class="q-answer"><p>Let P(n) be the proposition that 1 + 3 + 5 + ...+ 2n-1 = n² , <span class="math-tex">\(n\in \mathbb{N}\)</span></p><p>Show true for n=1,</p><p>LHS=2(1)-1=1</p><p>RHS=1²=1</p><p>LHS=RHS</p><p> <input type="text" style="height: auto;" data-c="Assume"> <span class="review"></span> true for n=k</p><p>1 + 3 + 5 + ...+ <input type="text" style="height: auto;" data-c="2k-1"> <span class="review"></span> = k²</p><p> <input type="text" style="height: auto;" data-c="Show"> <span class="review"></span> true for n=k+1</p><p>Show that 1 + 3 + 5 + ...+ 2k-1 + 2k+1 = <input type="text" style="height: auto;" data-c="(k+1)²"> <span class="review"></span> </p><p>LHS= 1 + 3 + 5 + ...+ 2k-1 + 2k+1</p><p>LHS= k² + <input type="text" style="height: auto;" data-c="2k+1"> <span class="review"></span> </p><p>LHS= (k+1)²</p><p>LHS= <input type="text" style="height: auto;" data-c="RHS"> <span class="review"></span> </p><p>True for <input type="text" style="height: auto;" data-c="n=k+1"> <span class="review"></span> </p><p>Concluding statement</p><p>True for n=1</p><p>Assuming it is true for n=k then it is true for n=k+1</p><p>Therefore it is true for all n , <span class="math-tex">\(n\in \mathbb{N}\)</span></p></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>Complete the gaps in the following proof by induction</p></div><div class="q-answer"><p>Let P(n) be the proposition that 9<sup>n</sup> - 1 is divisible by 8 , <span class="math-tex">\(n\in \mathbb{N}\)</span></p><p>Show true for n=1,</p><p>9<sup>1</sup> - 1 = 8 , 8 is divisible by 8</p><p>Hence true for n=1</p><p>Assume true for n=k</p><p><span class="math-tex">\(\frac{9^k-1}{8}=m\)</span> , <span class="math-tex">\(n\in \mathbb{Z}\)</span></p><p><span class="math-tex">\(9^k\)</span>= <input type="text" style="height: auto;" data-c="8m+1"> <span class="review"></span></p><p>Show true for n=k+1</p><p>9<sup>k+1</sup> - 1= <input type="text" style="height: auto;" data-c="9"> <span class="review"></span> 9<sup>k</sup> - 1</p><p>9<sup>k+1</sup> - 1=9(8m+1)-1</p><p>9<sup>k+1</sup> - 1=72m+ <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p>9<sup>k+1</sup> - 1=8( <input type="text" style="height: auto;" data-c="9m+1"> <span class="review"></span> )</p><p>Hence 9<sup>k+1</sup> - 1 is divisible by 8</p><p>True for n=k+1</p><p>Concluding statement</p><p>True for n=1</p><p>Assuming it is true for n=k then it is true for n= <input type="text" style="height: auto;" data-c="k+1"> <span class="review"></span> </p><p>Therefore it is true for all n , <span class="math-tex">\(n\in \mathbb{N}\)</span></p></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> </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> <div class="smart-object center" data-id="443"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that <span class="math-tex">\(12^n+2\times5^{n-1}\)</span> is divisible by 7 , <span class="math-tex">\(n\in\mathbb{Z^+}\)</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><content> </content>For assumption n = k is true</p> <p><span class="math-tex">\(12^k+2\times5^{k-1}=7m\)</span> , m is an integer</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/proof-by-induction/esq1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/esq1.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-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> <div class="smart-object center" data-id="444"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove by induction that <span class="math-tex">\(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =(n-1)2^ n+1 \ ,\ n\in\mathbb{Z^+}\)</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><content> </content>You might find it easier to if you write out the series in full:</p> <p><span class="math-tex">\(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =1\times2^0+3\times2^1+1\times2^2+...+n\times{ 2 }^{ n-1 }\)</span></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/proof-by-induction/esq2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/esq2.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="446"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL 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>Let <span class="math-tex">\(y = sinx\)</span></p> <p>Prove by induction that <span class="math-tex">\(\frac{d^ny}{dx^n}=sin(x+\frac{n\pi}{2})\)</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>The following trigonometric identity is useful for this proof:</p> <p><span class="math-tex">\(cosx \equiv sin(x+\frac{\pi}{2})\)</span></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/proof-by-induction/esq4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/esq4.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-default 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="445"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL 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>Prove by induction that <span class="math-tex">\(sinx+sin3x+sin5x+...+sin(2n-1)x=\frac{1-cos2nx}{2sinx} , \quad n\in\mathbb{Z^+} ,\quad sinx\neq 0\)</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>This is a challenging proof!</p> <p>The following identities will be useful:</p> <p><span class="math-tex">\(sin2x \equiv 2sinxcosx \\ cos2x \equiv 1-sin^2x\)</span></p> <hr> <p><span class="math-tex">\(sin(A+B) \equiv sinAcosB+cosAsinB\\ sin(2kx+x) \equiv sin2kxcosx+cos2kxsinx\)</span></p> <hr> <p><span class="math-tex">\(cos(A+B) \equiv cosAcosB+sinAsinB\\ cos(2kx+2x) \equiv cos2kxcos2x+sin2kxsin2x\)</span></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/proof-by-induction/esq3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/esq3.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="page-container panel-self-assessment" data-id="664"> <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>Proof by Induction</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" 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