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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">
<p>
A <strong>sequence</strong> is an ordered list of numbers. A <strong>series</strong> is the sum of
terms in a sequence. These ideas are key in algebra, finance, and modeling.
</p>
<h2 id="sequence-notation" class="book-prose__heading">Sequence notation</h2>
<p>
Terms are often written \(u_1,u_2,u_3,\dots\) where \(u_n\) is the \(n\)-th term.
</p>
<p><strong>Example:</strong> \(2,5,8,11,\dots\) has \(u_1=2\), \(u_2=5\), \(u_3=8\).</p>
<h2 id="arithmetic-sequences" class="book-prose__heading">Arithmetic sequences</h2>
<p>
Difference between consecutive terms is constant \(d\).
</p>
<p class="book-formula">\[
u_n=a+(n-1)d
\]</p>
<p>
where \(a\) is first term.
</p>
<p><strong>Example:</strong> \(a=4,d=3\Rightarrow u_n=4+3(n-1)=3n+1\).</p>
<h2 id="arithmetic-series" class="book-prose__heading">Arithmetic series</h2>
<p>
Sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)
\]</p>
<p>
where \(l\) is the \(n\)-th term.
</p>
<h2 id="geometric-sequences" class="book-prose__heading">Geometric sequences</h2>
<p>
Ratio between consecutive terms is constant \(r\).
</p>
<p class="book-formula">\[
u_n=ar^{n-1}
\]</p>
<p><strong>Example:</strong> \(3,6,12,24,\dots\) has \(a=3,r=2\).</p>
<h2 id="geometric-series" class="book-prose__heading">Geometric series</h2>
<p>
For \(r\ne1\), sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=a\frac{1-r^n}{1-r}
\]</p>
<p>
Equivalent form: \(S_n=a\frac{r^n-1}{r-1}\).
</p>
<h2 id="infinite-geometric" class="book-prose__heading">Infinite geometric series</h2>
<p>
If \(|r|&lt;1\), the sum to infinity exists:
</p>
<p class="book-formula">\[
S_\infty=\frac{a}{1-r}.
\]</p>
<p>
If \(|r|\ge1\), no finite sum to infinity.
</p>
<h2 id="sigma-notation" class="book-prose__heading">Sigma notation</h2>
<p>
\[
\sum_{k=1}^{n} f(k)
\]
means add values of \(f(k)\) from \(k=1\) to \(k=n\).
</p>
<p><strong>Example:</strong> \(\sum_{k=1}^{4}(2k+1)=3+5+7+9=24.\)</p>
<h2 id="problem-solving" class="book-prose__heading">Typical problem-solving workflow</h2>
<ol>
<li>Identify arithmetic or geometric behavior.</li>
<li>Find key parameters (\(a,d\) or \(a,r\)).</li>
<li>Use \(u_n\) for term questions, \(S_n\) for sum questions.</li>
<li>Check reasonableness (growth/decay, sign, size).</li>
</ol>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>Do not mix \(u_n\) and \(S_n\) formulas.</li>
<li>For geometric sums, check \(r\ne1\) before applying formula.</li>
<li>For infinity sums, always test \(|r|&lt;1\).</li>
<li>Define variables clearly before solving word problems.</li>
</ul>
</div>