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25bce65 @spicyj Sequences and series
spicyj authored
1 <!DOCTYPE html>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
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2 <html data-require="math math-format">
3 <head>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
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4 <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
25bce65 @spicyj Sequences and series
spicyj authored
5 <title>Geometric sequences 1</title>
ac1415e @spicyj Use requirejs for module loading
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6 <script data-main="../local-only/main.js" src="../local-only/require.js"></script>
25bce65 @spicyj Sequences and series
spicyj authored
7 </head>
8 <body>
9 <div class="exercise">
10 <div class="problems">
11 <div>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
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12 <div class="vars" data-ensure="abs(GIVEN[0][0]) &lt; 200 &amp;&amp; abs(GIVEN[0][1]) &lt; 200 &amp;&amp; abs(ANS[0]) &lt; 200 &amp;&amp; abs(ANS[1]) &lt; 200">
25bce65 @spicyj Sequences and series
spicyj authored
13 <var id="A">randRangeNonZero(-8, 8)</var>
14 <var id="RN">randFromArray([-1, 1]) * randRange(1, 4)</var>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
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15 <var id="RD" data-ensure="abs(RN) !== abs(RD)">randRange(1, 4)</var>
25bce65 @spicyj Sequences and series
spicyj authored
16 <var id="N">randRange(3, 5)</var>
17 <!-- N + 1 + OFFSET >= 0 always -->
18 <var id="OFFSET">randRange(-N - 1, 0)</var>
19 <var id="GIVEN">_.map(_.range(N), function(i) {
20 if (i + OFFSET &gt;= 0) {
21 return reduce(A * pow(RN, i + OFFSET), pow(RD, i + OFFSET));
22 } else {
23 return reduce(A * pow(RD, -i - OFFSET), pow(RN, -i - OFFSET));
24 }
25 })</var>
26 <var id="ANS">reduce(A * pow(RN, N + OFFSET), pow(RD, N + OFFSET))</var>
27
28 <var id="R_TEX">fractionReduce(RN, RD)</var>
29 <var id="GIVEN_TEX">_.map(GIVEN, function(f) {
30 return fractionReduce.apply(KhanUtil, f);
31 })</var>
32 </div>
33
34 <div class="question">
b0a7644 @csilvers Get rid of cardinal() everywhere escept the exercise testing cardinal…
csilvers authored
35 <p>The first <var>cardinalThrough20(N)</var> terms of a geometric sequence are given:</p>
25bce65 @spicyj Sequences and series
spicyj authored
36 <p><code><var>GIVEN_TEX.join(",")</var>, \ldots</code></p>
b0a7644 @csilvers Get rid of cardinal() everywhere escept the exercise testing cardinal…
csilvers authored
37 <p class="question">What is the <var>ordinalThrough20(N + 1)</var> term in the sequence?</p>
25bce65 @spicyj Sequences and series
spicyj authored
38 </div>
39 <div class="solution"><var>A * pow(RN / RD, N + OFFSET)</var></div>
40
41 <div class="hints">
42 <p>In any geometric sequence, each term is equal to the previous term times the common ratio.</p>
43 <p>Thus, the second term is equal to the first term times the common ratio. In this sequence, the second term, <code><var>GIVEN_TEX[1]</var></code>, is <code><var>R_TEX</var></code> times the first term, <code><var>GIVEN_TEX[0]</var></code>.</p>
44 <p>Therefore, the common ratio is <code><var>R_TEX</var></code>.</p>
b0a7644 @csilvers Get rid of cardinal() everywhere escept the exercise testing cardinal…
csilvers authored
45 <p>The <var>ordinalThrough20(N + 1)</var> term in the sequence is equal to the <var>ordinalThrough20(N)</var> term times the common ratio, or <code><var>GIVEN_TEX[N - 1]</var> \cdot <var>R_TEX</var> = <var>fractionReduce(A * pow(RN, N + OFFSET), pow(RD, N + OFFSET))</var></code>.</p>
25bce65 @spicyj Sequences and series
spicyj authored
46 </div>
47 </div>
48 </div>
49 </div>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
jeresig authored
50 </body>
51 </html>
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