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<!DOCTYPE html>
<html data-require="math math-format">
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<title>Arithmetic series</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
<div class="exercise">
<div class="problems">
<div id="explicit" data-calculator="">
<div class="vars" data-ensure="(D * (N - 1)) % 2 === 0 &amp;&amp; abs(SUM) &lt; 100">
<var id="A">randRangeNonZero(-40, 40)</var>
<var id="D">randFromArray([-1, 1]) * randRange(2, 6)</var>
<var id="N">randRange(5, 20)</var>
<var id="SUM">N * (2 * A + D * (N - 1)) / 2</var>
<div class="question">
<p>The arithmetic sequence <code>(a_i)</code> is defined by the formula:</p>
<p id="given-formula"><code>a_i = <var>A</var> + <var>D</var>(i - 1)</code></p>
<p class="question">
What is the sum of the first <code><var>N</var></code> terms in the series, starting with <code>a_1</code>?</p>
<div class="solution" data-forms="integer"><var>SUM</var></div>
<div class="hints">
<p>The sum of an arithmetic series is the number of terms in the series times the average of the first and last terms.</p>
<div id="recursive-hints" data-unwrap=""></div>
To find the sum of the first <code><var>N</var></code> terms,
we'll need the first and <var>ordinalThrough20(N)</var> terms of the series.
<p><code>a_1 = <var>A</var> + <var>D</var> (1 - 1) = <var>A</var></code></p>
<p><code>a_{<var>N</var>} = <var>A</var> + <var>D</var> (<var>N</var> - 1) = <var>A + D * (N - 1)</var></code></p>
<p>Therefore, the sum of the first <code><var>N</var></code> terms is</p>
<p><code>\qquad n\left(\dfrac{a_1 + a_{<var>N</var>}}{2}\right) = <var>N</var> \left(\dfrac{<var>A</var> + <var>A + D * (N - 1)</var>}{2}\right) = <var>SUM</var></code>.
<div id="recursive" data-calculator="" data-type="explicit">
<div class="question" data-apply="appendContents">
<p id="given-formula">
<code>a_1 = <var>A</var></code><br>
<code>a_i = a_{i-1} + <var>D</var></code><br>
<div class="hints" data-apply="appendContents">
<div id="recursive-hints" data-unwrap="">
<p>First, let's find the explicit formula for the terms of the arithmetic series. We can see that the first term is <code><var>A</var></code> and the common difference is <code><var>D</var></code>.</p>
<p>Thus, the explicit formula for this sequence is <code>a_i = <var>A</var> + <var>D</var>(i - 1)</code>.</p>