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Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

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<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Sequences and sums appear in ancient mathematics, including arithmetic progressions used for
counting and trade. Later, geometric series became central in finance and calculus.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="derive-arithmetic-sum" class="book-prose__heading">Why the arithmetic sum formula works</h3>
<p>
Write \(S_n=a+(a+d)+\dots+l\) and reverse it. Adding line-by-line gives \(2S_n=n(a+l)\), so
\(S_n=\frac{n}{2}(a+l)\).
</p>
<h3 id="derive-geometric-sum" class="book-prose__heading">Why the geometric sum formula works</h3>
<p>
Let \(S_n=a+ar+\dots+ar^{n-1}\). Multiply by \(r\), subtract, and simplify:
\[
(1-r)S_n=a(1-r^n).
\]
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">Find the 20th term of \(5,8,11,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_{20}=5+19(3)=62\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Find \(S_{30}\) for \(2,5,8,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=2,d=3\Rightarrow S_{30}=\frac{30}{2}[2(2)+29(3)]=1365\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">For \(3,6,12,\dots\), find the 8th term.</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_8=3\cdot2^7=384\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Compute \(1+\frac12+\frac14+\frac18+\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=1,r=\frac12\Rightarrow S_\infty=\frac{1}{1-1/2}=2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Evaluate \(\sum_{k=1}^{5}(k+2)\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(3+4+5+6+7=25\).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Savings plans:</strong> recurring deposits form sequences and cumulative sums.</li>
<li><strong>Population models:</strong> repeated growth can be modeled geometrically.</li>
<li><strong>Computer science:</strong> loops and algorithm costs often use summation notation.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school algebra curricula on arithmetic and geometric progressions.</li>
<li>Precalculus texts on sigma notation and finite/infinite geometric series.</li>
</ul>
</div>