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<div id="main-column" class="span9"> <article id="1a-sequences-series-intro-sn" style="margin-top: 16px;">
<h1 class="section-title">1A. Sequences & series intro (SN)</h1>
<ul class="breadcrumb"><li><a title="Home" href="../../../index.html"><i class="fa fa-home"></i></a><span class="divider">/</span></li><li><span class="gray">1. Number & Algebra</span><span class="divider">/</span></li><li><span class="active">1A. Sequences & series intro (SN)</span></li></ul>
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<h3><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/seq-series-img0-1.jpg" style="float: right;"><span style="color:#0000FF;"></span>∼ Student Notes ∼</h3><h3>Section1A. Sequences & series introduction</h3><p style="margin-left: 40px;"><strong>Sub-sections on this page:</strong><span style="color:#000000;"><br><a class="scroll-to" data-target="Basics"><u>1A.1</u> Basics</a></span><span style="color:#000000;"><br><a class="scroll-to" data-target="term"><u>1A.2</u> Finding the <span class="math-tex">\(n\)</span><sup>th</sup> term</a></span><span style="color:#000000;"><br><a class="scroll-to" data-target="tech"><u>1A.3</u> Using technology</a></span><a class="scroll-to" data-target="summary"><span style="color:#000000;"></span></a><br><span style="color:#000000;"></span><span style="color:#000000;"></span><a class="scroll-to" data-target="summary"><u>1A.4</u> Summary</a><br><a class="scroll-to" data-target="challenge"><u>1A.5</u> Challenge question</a></p><hr><h4><strong><a class="anchor" id="Basics" name="Basics"> </a><span style="color:#000000;">1A.1 Basics</span></strong></h4><p>A numerical <strong>sequence</strong> is a set of numbers listed in a specific order that is determined by a rule. For example, the sequence <span class="math-tex">\(3,\;7,\;11,\;15,\;19,\;23\)</span> is a list of six numbers that begins with the number 3 and the remaining numbers in the list are determined by applying the rule “add 4” to proceed from one number to the next in the list. A sequence is sometimes also referred to as a <strong>progression</strong>.</p><div class="pinkBg"><p>A <strong>sequence </strong>is a set (or list) of numbers, separated by commas, in which each number after the first is determined by a rule.</p></div><p>Each number in a sequence is a called a <strong>term</strong> of the sequence. A term is referred to by its position in the sequence; for example: first term, 3<sup>rd</sup> term, 10<sup>th</sup> term, last term.</p><p>It is very helpful to use a variable (letter) to represent the terms of a sequence. The variable <em><span class="math-tex">\(u\)</span></em> is most often used, but others such as <span class="math-tex">\(a\)</span> and <span class="math-tex">\(t\)</span> are also common. Individual terms are identified using <strong>subscript notation</strong> (also referred to as suffix notation) where a small number (the subscript or suffix) is written below and to the right of the sequence variable to indicate the position of the term in the sequence. For example, in the sequence <span class="math-tex">\(3,\;7,\;11,\;15,\;19,\;23\)</span> the fact that the “3<sup>rd</sup> term is 11” is written as <span class="math-tex">\({u_3} = 11\)</span>. The subscript can also be represented by a variable (usually <span class="math-tex">\(n\)</span> or <span class="math-tex">\(r\)</span>, but other letters such as <span class="math-tex">\(i\)</span> are also used) so that the <em><span class="math-tex">\(n\)</span></em><sup>th</sup> term of a sequence (i.e. the <em>general</em> term) is expressed as <span class="math-tex">\({u_n}\)</span>.</p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/nmbr-terms-q-img2.jpg" style="border-width: 0px; border-style: solid; float: right; width: 275px; height: 116px;">Consider the following sequences.</p><p>(a) <span class="math-tex">\(1,\;4,\;9,\;16,\;25\)</span></p><p>(b) <span class="math-tex">\(3,\;7,\;11,\;15,\; \ldots ,\;43,\;47\)</span></p><p>(c) <span class="math-tex">\(2,\;1,\;\frac{1}{2},\;\frac{1}{4},\;\frac{1}{8}\)</span></p><p>(d) <span class="math-tex">\(1,\;1,\;2,\;3,\;5,\;8,\;13,\; \ldots \)</span></p><p>The sequences (b) and (d) make use of three dots (…) to indicate that omitted numbers continue in the same pattern. The three dots symbol is called an <strong>ellipsis</strong> (not to be confused with ellipse). In a sequence, three dots can be used between two terms of a sequence such as in (b), or at the end of a sequence as in (d). An ellipsis appearing at the end of a sequence indicates that the sequence continues without end. Such a sequence has an infinite number of terms and, hence, is called an <strong>infinite</strong> <strong>sequence</strong>. The sequences (a), (b) and (c) end after a countable, or finite, number of terms; so, each of these is a <strong>finite sequence</strong>. The sequences (a) and (c) each have five terms and even though it may not be immediately clear how many terms are in sequence (b), the sequence definitely ends (it’s not difficult to work out that it has 12 terms).</p><p>The sum of a sequence of numbers is called a <strong>series</strong>. For example, if the terms in sequence (c) are added together we get the series <span class="math-tex">\(2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\)</span>.</p><div class="pinkBg"><p>When the terms of a sequence are added together the sum of the terms is called a <strong>series</strong>.</p></div><p>Given that this series only has five terms, it is not difficult to manually calculate that the value of the series is <span class="math-tex">\(\frac{{31}}{8}\)</span>. If the series had 20 terms, it would be useful to have a formula to compute the sum rather than attempting to do it manually. Although the sequence <span class="math-tex">\(2,\;1,\;\frac{1}{2},\;\frac{1}{4},\;\frac{1}{8}\)</span> and the series <span class="math-tex">\(2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\)</span> are closely related (their terms are the same), it is important to note that a sequence is a <em>list of numbers</em> whereas a series is a <em><strong>sum</strong> of a list of numbers</em>. The subscripted symbol <span class="math-tex">\({S_n}\)</span> is used to denote the sum of the first <span class="math-tex">\(n\)</span> terms of a series, so that <span class="math-tex">\({S_n} = {u_1} + {u_2} + {u_3} + \; \cdots \; + {u_n}\)</span>. It's important to note that in this case <span class="math-tex">\(n\)</span> is a fixed number (constant) for a particular sum since it indicates how many terms are to be added together (a letter other than <span class="math-tex">\(n\)</span> could be used).</p><div class="row-fluid"><div class="span6 col-left"><div><hr class="hidden"><p>The sum of the first <span class="math-tex">\(n\)</span> terms in a series with the general term <span class="math-tex">\({u_r}\)</span> (<span class="math-tex">\(r\)</span><sup>th</sup> term) where <span class="math-tex">\(r\)</span> is a variable (a 'counter' that starts at 1 and goes to <span class="math-tex">\(n\)</span>) is most efficiently expressed using <strong>sigma notation</strong> <span class="math-tex">\(\sum\limits_{r = 1}^n {{u_r}} = {u_1} + {u_2} + {u_3} + \; \cdots \; + {u_n}\)</span>. Sigma notation is an efficient way to represent a sum using the symbol <span class="math-tex">\(\sum \)</span> (Greek capital letter <em>sigma</em>).</p></div></div><div class="span6"><div><p style="text-align: center;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/sigma-img0.jpg" style="width: 280px; height: 176px;"></p></div></div></div><div class="greenBg"><p>Note that the variable <span class="math-tex">\(r\)</span> does not appear when <span class="math-tex">\(\sum\limits_{r = 1}^n {{u_r}} \)</span> is written out: <span class="math-tex">\(\sum\limits_{r = 1}^n {{u_r}} = {u_1} + {u_2} + {u_3} + \; \cdots \; + {u_n}\)</span><br>A letter other than <span class="math-tex">\(r\)</span> can be the variable. For example, <span class="math-tex">\(\sum\limits_{i = 1}^n {{u_i}} = {u_1} + {u_2} + {u_3} + \; \cdots \; + {u_n}\)</span> or <span class="math-tex">\(\sum\limits_{n = 1}^4 {{u_n}} = {u_1} + {u_2} + {u_3} + {u_4}\)</span><br>Also, as mentioned, the terms can be represented by a letter other than <span class="math-tex">\(u\)</span>. For example, <span class="math-tex">\(\sum\limits_{r = 1}^5 {{a_r}} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5}\)</span> or <span class="math-tex">\(\sum\limits_{k = 1}^n {{t_k}} = {t_1} + {t_2} + {t_3} + \cdots + {t_n}\)</span></p></div><p>Sequences and series are classified by whether they are <strong>finite </strong>or <strong>infinite</strong>; and can also be classified by the rule which generates the terms in the sequence. Sequence (b) is an <strong>arithmetic sequence</strong> because each term is obtained by <u>adding</u> a constant (can be positive or negative), called the <strong>common difference <em><span class="math-tex">\(d\)</span></em></strong>, to the previous term; <span class="math-tex">\(d = 4\)</span> for sequence (b). Each term in sequence (c) is obtained by <u>multiplying</u> a constant, called the <strong>common ratio <em><span class="math-tex">\(r\)</span></em></strong>, to the previous term; <span class="math-tex">\(r = \frac{1}{2}\)</span> for sequence (c). This type of sequence is called a <strong>geometric sequence</strong>.<img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/seq-intro-img0.jpg" style="float: right; width: 250px; height: 142px;"></p><div class="pinkBg"><p>The only types of <strong>numerical sequences and series</strong> specifically included in the <strong>course syllabus</strong> are arithmetic and geometric. Nevertheless, in these introductory notes it is helpful to briefly consider some other sequences that are neither arithmetic nor geometric in order to illustrate some concepts regarding sequences.</p></div><p>Although sequences (a) and (d) are neither arithmetic nor geometric, each has a clear rule for generating its terms (more on that in the next section). Sequences that are particularly noteworthy are often given a special name; for example, you may recognize sequence (d) as the <strong>Fibonacci sequence</strong>. The <a href="https://oeis.org/" target="_blank">On-Line Encyclopedia of Integer Sequences</a> (a searchable database containing over 335,000 sequences started in 1964) includes the Fibonacci sequence amongst its list of seven ‘famous sequences’.</p><hr><h4><strong><a class="anchor" id="term" name="term"> </a><span style="color:#000000;">1A.2 Finding the <em>n</em><sup>th</sup> term (general term)</span></strong></h4><p>Another well-known sequence is the <strong>triangular numbers</strong> which can be illustrated with a triangular dot pattern as shown below.</p><div class="polaroid-center"><img src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/tri_nmbrs2_img.jpg" style="margin: 8px 0px; width: 440px; height: 114px;"><div class="caption">The first six terms of the <em>triangular numbers</em> sequence.</div></div><p>If <span class="math-tex">\({t_n}\)</span> represents the <span class="math-tex">\(n\)</span><sup>th</sup> triangular number (i.e. the <strong>general term</strong> of the sequence), it is clear that:</p><p><span class="math-tex">\({t_1} = 1\)</span></p><p><span class="math-tex">\({t_2} = {t_1} + 1 = 1 + 2 = 3\)</span></p><p><span class="math-tex">\({t_3} = {t_2} + 3 = 3 + 3 = 6\)</span></p><p><span class="math-tex">\({t_4} = {t_3} + 4 = 6 + 4 = 10\)</span> ; and so on.</p><div class="row-fluid"><div class="span6 col-left"><div><hr class="hidden"><p>This pattern leads to the following formula for the <span class="math-tex">\(n\)</span><sup>th</sup> triangular number: <span class="math-tex">\({t_1} = 1\)</span>, <span class="math-tex">\({t_n} = {t_{n - 1}} + n\)</span><br>This formula says “The first term is 1, and the <span class="math-tex">\(n\)</span><sup>th</sup> term is equal to the previous term plus <span class="math-tex">\(n\)</span>." This type of formula is called <strong>recursive</strong> because it describes how later terms in the sequence are computed in terms of previous terms. Knowing that the 6<sup>th</sup> triangular number is 21, we can easily see that the 7<sup>th</sup> triangular number is 28 <span class="math-tex">\(\left( {21 + 7 = 28} \right)\)</span>. But, what if we want to know the 30<sup>th</sup> triangular number? A <strong>recursive formula</strong> is not convenient for computing terms further on in a sequence. Spreadsheets are a handy tool for computing multiple terms in a sequence. Play the video (no sound) at right to see <strong>Excel </strong>being used to find the 30th triangular number. In the next sub-section we'll take a brief look at using GDCs to work with sequences.</p></div></div><div class="span6"><div><p style="text-align: center;"><iframe frameborder="0" height="360" scrolling="no" src="https://player.vimeo.com/video/435880461" width="150"></iframe></p></div></div></div><div class="pinkBg"><p>A <strong>recursive formula</strong> is also referred to as an <strong>inductive formula</strong> or a <strong>recurrence relation</strong>. Although this kind of formula - that computes terms in a sequence based on previous term(s) - is helpful for defining some sequences, it is <u>not</u> in the syllabus for this course.</p></div><p>It is much more convenient to have a formula for the <span class="math-tex">\(n\)</span><sup>th</sup> term in a sequence that is expressed <em>only</em> in terms of <span class="math-tex">\(n\)</span> (the number, or position, of the term) and does not rely on knowing previous terms. This type of formula is usually called an <strong>explicit formula</strong> (or closed formula) where he <span class="math-tex">\(n\)</span><sup>th</sup> term is written as a function of the independent variable <span class="math-tex">\(n\)</span> (or whichever variable represents the number of the term). In the next two sections (<strong>1.B</strong> and <strong>1.C</strong>) we will see that finding an explicit formula for the general term in an arithmetic or geometric sequence is a straightforward process (they are in the formula booklet). Finding the explicit formula for the general term in other sequences is often less straightforward. What about an explicit formula for the <span class="math-tex">\(n\)</span><sup>th</sup> triangular number?</p><p>Instead of displaying the dot pattern for triangular numbers as shown above, let's change the shape of the triangular pattern ...</p><p style="text-align: center;"><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/tri_nmbrs3_img.jpg" style="width: 500px; height: 119px;"></p><p>... and then create a duplicate of each triangle (in red) and place it along the existing triangle to form a rectangular dot pattern.</p><p style="text-align: center;"><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/tri_nmbrs4_img.jpg" style="width: 500px; height: 121px;"></p><p>The black dots make up half of the total number of dots in each 'rectangle'. This shows that the <strong>explicit formula</strong> for the <span class="math-tex">\(n\)</span><sup>th</sup> triangular number is <span class="math-tex">\({t_n} = \frac{1}{2}n\left( {n + 1} \right)\)</span>. No need for a spreadsheet now ... the 30<sup>th</sup> triangular number can be computed quickly by simply substituting <span class="math-tex">\(n = 30\)</span> into the formula: <span class="math-tex">\({t_{30}} = \frac{1}{2} \cdot 30\left( {30 + 1} \right) = 15 \cdot 31 = 465\)</span></p><p>Consider again sequences (a) <span class="math-tex">\(1,\;4,\;9,\;16,\;25\)</span> and (d) <span class="math-tex">\(1,\;1,\;2,\;3,\;5,\;8,\;13,\; \ldots \)</span> (Fibonacci sequence) from above. Let <span class="math-tex">\({u_n}\)</span> represent the <span class="math-tex">\(n\)</span><sup>th</sup> term in sequence (a) and let <span class="math-tex">\({F_n}\)</span> be the <span class="math-tex">\(n\)</span><sup>th</sup> Fibonacci number. The explicit formula for sequence (a) is obviously <span class="math-tex">\({u_n} = {n^2}\)</span>. An explicit formula does exist for the Fibonacci sequence (<a href="https://artofproblemsolving.com/wiki/index.php/Binet%27s_Formula" target="_blank"><u>Binet's formula</u></a>) but it is quite complicated and difficult to derive (definitely not obvious). However, it is not difficult to determine a recursive formula for the Fibonacci sequence. Instead of starting with the value of one term - as we did for the triangular numbers - a recursive formula for the Fibonacci requires starting with the value of the first <em>two</em> terms since the rule for Fibonacci is that each term (from the 3<sup>rd</sup> term onwards) is the sum of the previous <em>two</em> terms. Hence, the recursive formula for the Fibonacci sequence is <span class="math-tex">\({F_1} = 1,\;\;{F_2} = 1,\;\;{F_n} = {F_{n - 2}} + {F_{n - 1}}\)</span> for <span class="math-tex">\(n > 2\)</span> (typically <span class="math-tex">\({F_0} = 0\)</span>).</p><p>In general, it's much more useful to have an explicit formula for finding the <span class="math-tex">\(n\)</span><sup>th</sup> term of a sequence especially if you do not have access to technology. However, a recursive formula is usually easier to find - and, if you do have access to technology (next sub-section) it is no more difficult to use a recursive formula than an explicit formula for finding terms of a sequence.</p><hr><h4><strong><a class="anchor" id="tech" name="tech"> </a><span style="color:#000000;">1A.3 Using technology</span></strong></h4><p>Technology such as a GDC or spreadsheet software are superb tools for generating and exploring sequences. Be sure to be familiar with sequence commands and operations on the GDC model you are using. Below are images and videos (no sound) showing how one could perform two tasks using two different calculators: the <strong>TI-84 Plus</strong> and the <strong>TI-Nspire CX</strong>. In sub-section <strong>1A.2</strong> above, a video showed <strong>Excel </strong>being used to find the 30<sup>th</sup> term in the triangular numbers sequence.</p><p><strong><u>Task #1</u></strong> is simply to generate two sequences: <strong>(i)</strong> sequence (a) <span class="math-tex">\(1,\;4,\;9,\;16,\;25\)</span> with the general term given by the explicit formula <span class="math-tex">\({u_n} = {n^2}\)</span> (sequence (a) is neither arithmetic nor geometric); and, <strong>(ii)</strong> the geometric sequence <span class="math-tex">\(2,\;1,\;\frac{1}{2},\;\frac{1}{4},\;\frac{1}{8}\)</span> (sequence (c) from above). We will study explicit formulas for the <span class="math-tex">\(n\)</span><sup>th</sup> term of a geometric sequence in Section <strong>1.C</strong>; so, a recursive formula will be used here (it's easy to see that the initial term is 2 and afterwards each term is obtained by multiplying the previous term by <span class="math-tex">\(\frac{1}{2}\)</span>).</p><p><u><strong>Task #2</strong></u> is to investigate the ratio of successive terms in the Fibonacci sequence, <span class="math-tex">\(\frac{{{F_n}}}{{{F_{n - 1}}}}\)</span>, as <span class="math-tex">\(n\)</span> increases (using the first 30 Fibonacci numbers).</p><h5><span lang="EN-US" style="font-size:12.0pt;line-height:
115%;font-family:"Times New Roman",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-SA">■</span> TI-84 Plus <span lang="EN-US" style="font-size:12.0pt;line-height:
115%;font-family:"Times New Roman",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-SA">■</span></h5><p><strong>Task #1 <br>(i): </strong>SInce we know an explicit formula for sequence (a) then we can use the sequence command (<strong>seq</strong>). Press '2nd', 'stat', 'right arrow' to the <strong>LIST OPS</strong> menu. Select the the sequence command (<strong>5. seq</strong>), enter the sequence's characteristics (expression, variable, begin, end), then 'paste', and the command can then be performed on the home screen displaying the terms of the sequence as a list.</p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/84-seq-list-img.jpg" style="border-width: 1px; border-style: solid; width: 130px; height: 110px;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/84-seq-wiz-img.jpg" style="border-width: 1px; border-style: solid; width: 126px; height: 109px; margin: 0px 20px;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/ti84_seqa_img.jpg" style="width: 320px; height: 91px; margin: 0px 15px; border-width: 1px; border-style: solid;"></p><p><strong>(ii)</strong> Since we're applying a recursive formula to create sequence (c), we use sequence graphing mode (<strong>SEQ</strong>) to enter the recursive formula as the function <span class="math-tex">\(u\left( n \right)\)</span>. Press 'mode' then move down to the 5th line and change the graphing mode from <strong>FUNCTION </strong>to <strong>SEQ</strong>. Press 'Y=' and enter the recursive formula <span class="math-tex">\(u\left( n \right) = \frac{1}{2}{u_{n - 1}}\)</span>. It is not possible to enter subscripts on the TI-84, so <span class="math-tex">\({u_{n - 1}}\)</span> must be entered as <span class="math-tex">\(u\left( {n - 1} \right)\)</span>. The function name <span class="math-tex">\(u\)</span> is obtained by pressing '2nd', '7'; and the sequence variable <span class="math-tex">\(n\)</span> is obtained by pressing 'X,T,<span class="math-tex">\(\theta \)</span>,<span class="math-tex">\(n\)</span>'. Now create the sequence using the <strong>seq</strong> command (as above for sequence (a)) - entering <span class="math-tex">\(u\left( n \right)\)</span> for the expression and <span class="math-tex">\(n\)</span> for the variable.</p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/84-mode-img.jpg" style="float: left; width: 250px; height: 121px; border-width: 1px; border-style: solid;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/ti84_geo1_img1.jpg" style="width: 293px; height: 140px; border-width: 1px; border-style: solid;"></p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/84-seqc-wiz-img.jpg" style="float: left; width: 150px; height: 130px; border-width: 1px; border-style: solid;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/ti84_geo1_img2.jpg" style="border-width: 1px; border-style: solid; width: 320px; height: 120px;"></p><hr><p><strong>Task #2</strong> video (no sound)</p><p><iframe frameborder="0" height="351" scrolling="no" src="https://player.vimeo.com/video/436113420" width="624"></iframe></p><hr><h5><span lang="EN-US" style="font-size:12.0pt;line-height:
115%;font-family:"Times New Roman",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-SA">■</span> TI-Nspire <span lang="EN-US" style="font-size:12.0pt;line-height:
115%;font-family:"Times New Roman",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-SA">■</span></h5><p><strong>Task #1<br>(i) </strong>Enter the explicit formula for sequence (a) by using the sequence command (<strong>seq</strong>). It is best to access <strong>seq </strong>command from the <strong>Catalog </strong>(button has image of a book on it and is directly below the delete button). Highlighting <strong>seq</strong> in the list of commands in the <strong>Catalog </strong>displays the required syntax (at bottom) for the command: expression, variable, lowest value of variable, highest value of variable (same as TI-84; 'Step' is optional since it is in square brackets).<br><strong>(ii)</strong> To enter the recursive formula for sequence (c), use the <strong>seqn</strong> command found directly below the <strong>seq</strong> command in the <strong>Catalog</strong>. Enter the recursive formula <span class="math-tex">\(\frac{1}{2}{u_{n - 1}}\)</span> for the expression of <span class="math-tex">\(u\)</span> in terms of <span class="math-tex">\(n\)</span>, then list of initial terms. There is only one initial term, i.e. <span class="math-tex">\({u_1} = 2\)</span> which is contained in curly brackets because it is a list. Enter 5 for the maximum value of <span class="math-tex">\(n\)</span>.</p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/nspire-ctlg-seq2-img.jpg" style="border-width: 1px; border-style: solid; margin: 0px 15px; width: 200px; height: 106px;"><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/nspire-ctlg-seqn-img.jpg" style="border-width: 1px; border-style: solid; margin: 0px 15px; width: 250px; height: 81px;"></p><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/nspire_seqac_img.jpg" style="border-width: 1px; border-style: solid; margin: 2px 15px; width: 360px; height: 144px;"></p><hr><p><strong>Task #2</strong> video (no sound)</p><p><strong><iframe frameborder="0" height="351" scrolling="no" src="https://player.vimeo.com/video/436255791" width="624"></iframe></strong></p><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="summary" name="summary"> </a></strong></span><span style="color:#000000;"><strong>1A.4 Summary</strong></span><span style="color:#0000FF;"></span></h4><ul><li>A numerical <strong>sequence</strong> is a set of numbers listed in a specific order determined by a rule.</li><li>Each number in a sequence is called a <strong>term</strong> of the sequence and is identified using <strong>subscript notation</strong>; e.g. <span class="math-tex">\({u_5}\)</span> refers to the fifth term.</li><li>Sequences either have a countable number of terms (<strong>finite sequence</strong>) or the terms of a sequence continue indefinitely (<strong>infinite sequence</strong>).</li><li>A <strong>series </strong>is formed by adding all the terms of a sequence. <u>Note</u>: a finite series will always have a sum whereas an infinite series may or may not converge to a sum (covered in Section <strong>1.C</strong> on geometric sequences and series).</li><li><strong>Sigma notation</strong>: Greek letter <span class="math-tex">\(\sum \)</span> (sigma) is used to indicate a sum; e.g. <span class="math-tex">\(\sum\limits_{r = 1}^n {{u_r}} = {u_1} + {u_2} + \; \cdots \; + {u_n}\)</span> where <span class="math-tex">\(n\)</span> is a constant and <span class="math-tex">\(r\)</span> is a variable (counter) that goes from 1 to <span class="math-tex">\(n\)</span>. <u>Note</u>: letters other than <span class="math-tex">\(n\)</span> and <span class="math-tex">\(r\)</span> may be used</li><li>An <strong>arithmetic sequence</strong> is one such that each term is obtained by <u>adding</u> a constant (<strong>common difference <em><span class="math-tex">\(d\)</span></em></strong>) to the previous term (see Section <strong>1.B</strong>).</li><li>A <strong>geometric sequence</strong> is one such that each term is obtained by <u>multiplying</u> a constant (<strong>common ratio <em><span class="math-tex">\(r\)</span></em></strong>) to the previous term (see Section <strong>1.C</strong>). <u>Note</u>: arithmetic and geometric are the only type of sequences/series included in the syllabus</li><li>A formula for the <span class="math-tex">\(n\)</span><sup>th</sup> term in a sequence (i.e. the <strong>general term</strong>) can be <strong>recursive</strong> or <strong>explicit</strong>. <u>Note</u>: recursive formulas are not included in the course syllabus</li><li>A <strong>recursive </strong>formula for a sequence provides an initial term (or terms) and a rule for computing subsequent terms from the previous term (or terms).</li><li>An <strong>explicit </strong>formula for a sequence is a rule used to compute each term directly from only the number (position) of the term.</li><li>Be familiar with sequence commands and operations on the <strong>GDC</strong> model you are using. <strong>Spreadsheet </strong>software is very useful for constructing sequences and could be helpful if you were to investigate a sequence as part of your Exploration (IA).</li></ul><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="challenge" name="challenge"> </a></strong></span><span style="color:#000000;"><strong>1A.5 Challenge question</strong></span></h4><p><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/challeng-q-img.jpg" style="width: 350px; height: 80px;"></p><section class="tib-hiddenbox"><p><u>answer</u>: <strong>1010</strong></p><p>Each number in the sequence has a value of 10 written in different bases. The first number is in base 10, the next in base 9, the next in base 8, and so on. 1010 has a value of 10 in base 2.<br><img alt="" src="../../../ib/mathanalysis/analysis/number---algebra/seq-series/ch-ans-img2.jpg" style="width: 320px; height: 27px;"></p></section><hr><table width="100%"><tbody><tr><td> <span style="color:#B22222;"></span><br> <a href="1a-sequences-series-intro-sn.html" target="_self"><span style="color:#B22222;"></span></a></td><td style="text-align: right;"><span style="color:#B22222;"></span> go to next section: <a href="../34896/1b-arithmetic-sequences-series-sn.html" target="_self">1B. Arithmetic sequences & series</a> → </td></tr></tbody></table><script>document.querySelectorAll('.tib-teacher-only').forEach(e => e.remove());</script>
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