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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Geometric sequences 1</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<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">
<var id="A">randRangeNonZero(-8, 8)</var>
<var id="RN">randFromArray([-1, 1]) * randRange(1, 4)</var>
<var id="RD" data-ensure="abs(RN) !== abs(RD)">randRange(1, 4)</var>
<var id="N">randRange(3, 5)</var>
<!-- N + 1 + OFFSET >= 0 always -->
<var id="OFFSET">randRange(-N - 1, 0)</var>
<var id="GIVEN">_.map(_.range(N), function(i) {
if (i + OFFSET &gt;= 0) {
return reduce(A * pow(RN, i + OFFSET), pow(RD, i + OFFSET));
} else {
return reduce(A * pow(RD, -i - OFFSET), pow(RN, -i - OFFSET));
}
})</var>
<var id="ANS">reduce(A * pow(RN, N + OFFSET), pow(RD, N + OFFSET))</var>
<var id="R_TEX">fractionReduce(RN, RD)</var>
<var id="GIVEN_TEX">_.map(GIVEN, function(f) {
return fractionReduce.apply(KhanUtil, f);
})</var>
</div>
<div class="question">
<p>The first <var>cardinal(N)</var> terms of a geometric sequence are given:</p>
<p><code><var>GIVEN_TEX.join(",")</var>, \ldots</code></p>
<p class="question">What is the <var>ordinal(N + 1)</var> term in the sequence?</p>
</div>
<div class="solution"><var>A * pow(RN / RD, N + OFFSET)</var></div>
<div class="hints">
<p>In any geometric sequence, each term is equal to the previous term times the common ratio.</p>
<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>
<p>Therefore, the common ratio is <code><var>R_TEX</var></code>.</p>
<p>The <var>ordinal(N + 1)</var> term in the sequence is equal to the <var>ordinal(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>
</div>
</div>
</div>
</div>
</body>
</html>
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