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Tutoring abilities

Imagine receiving $2 on Monday, $6 on Tuesday, $18 on Wednesday, and $54 on Thursday. If you want to know the total amount in your pocket by Thursday night, you are working with a **geometric series**.

A sequence is an ordered list of numbers, whereas a **series** is the indicated sum of the terms in a sequence.

In a **geometric sequence**, each term is generated by multiplying the preceding term by a fixed non-zero value called the **common ratio** ($r$).

When we list the payouts separated by commas ($2, 6, 18, 54$), we have a sequence. When we add them together ($2 + 6 + 18 + 54$), we form a series.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .nav-toggle {
      display: flex;
      background: #e2e8f0;
      padding: 3px;
      border-radius: 10px;
      gap: 4px;
      margin-bottom: 12px;
    }

    .toggle-btn {
      border: none;
      background: transparent;
      padding: 6px 16px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .toggle-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 1px 3px rgba(0,0,0,0.1);
    }

    .display-area {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
      flex-grow: 1;
      justify-content: center;
    }

    /* Term row */
    .terms-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      width: 100%;
    }

    .term-box {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      border-radius: 10px;
      font-size: 18px;
      font-weight: 700;
      color: #1e2945;
      width: 48px;
      height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: all 0.3s ease;
      position: relative;
    }

    .term-box.active {
      border-color: #0284c7;
      background: #f0f9ff;
      color: #0284c7;
      box-shadow: 0 4px 12px rgba(2, 132, 199, 0.15);
    }

    .term-box .mult-badge {
      position: absolute;
      top: -24px;
      font-size: 11px;
      font-weight: 700;
      color: #0284c7;
      background: #e0f2fe;
      padding: 1px 6px;
      border-radius: 6px;
      opacity: 0;
      transition: opacity 0.3s ease;
    }

    .show-mult .mult-badge {
      opacity: 1;
    }

    .separator {
      font-size: 20px;
      font-weight: 700;
      color: #94a3b8;
      width: 16px;
      text-align: center;
      transition: all 0.2s ease;
    }

    /* Stacked Cumulative Bar */
    .bar-container {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .bar-track {
      height: 28px;
      background: #f1f5f9;
      border-radius: 6px;
      display: flex;
      overflow: hidden;
      width: 100%;
    }

    .bar-seg {
      height: 100%;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 12px;
      font-weight: 700;
      color: #ffffff;
      transition: width 0.4s ease, opacity 0.3s ease;
      opacity: 0.25;
    }

    .bar-seg.lit {
      opacity: 1;
    }

    .seg-1 { background: #38bdf8; width: 2.5%; }
    .seg-2 { background: #0284c7; width: 7.5%; }
    .seg-3 { background: #0369a1; width: 22.5%; }
    .seg-4 { background: #0c4a6e; width: 67.5%; }

    .sum-readout {
      display: flex;
      justify-content: space-between;
      align-items: baseline;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
    }

    .sum-val {
      font-size: 20px;
      font-weight: 800;
      color: #0284c7;
    }

    /* Stepper Controls */
    .stepper {
      display: flex;
      align-items: center;
      gap: 8px;
    }

    .step-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .step-btn:active {
      background: #f1f5f9;
    }

    .step-btn:disabled {
      opacity: 0.35;
      cursor: not-allowed;
    }

    .step-dots {
      display: flex;
      gap: 6px;
    }

    .dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      transition: background 0.2s ease;
    }

    .dot.active {
      background: #0284c7;
      transform: scale(1.2);
    }
  </style>
</head>
<body>

  
  <div class="nav-toggle">
    <button class="toggle-btn active" id="btnSequence">Sequence</button>
    <button class="toggle-btn" id="btnSeries">Series</button>
  </div>

  
  <div class="display-area" id="diagramArea">
    
    <div class="terms-row show-mult" id="termsRow">
      <div class="term-box active" id="box1">2</div>
      <div class="separator" id="sep1">,</div>
      <div class="term-box active" id="box2">
        <span class="mult-badge">×3</span>6
      </div>
      <div class="separator" id="sep2">,</div>
      <div class="term-box active" id="box3">
        <span class="mult-badge">×3</span>18
      </div>
      <div class="separator" id="sep3">,</div>
      <div class="term-box active" id="box4">
        <span class="mult-badge">×3</span>54
      </div>
    </div>

    
    <div class="bar-container">
      <div class="bar-track">
        <div class="bar-seg seg-1 lit" id="seg1"></div>
        <div class="bar-seg seg-2 lit" id="seg2"></div>
        <div class="bar-seg seg-3 lit" id="seg3"></div>
        <div class="bar-seg seg-4 lit" id="seg4"></div>
      </div>
      <div class="sum-readout">
        <span id="sumLabel">List of Terms</span>
        <span class="sum-val" id="sumDisplay">4 terms</span>
      </div>
    </div>
  </div>

  
  <div class="stepper" id="stepperControls" style="visibility: hidden;">
    <button class="step-btn" id="prevBtn">←</button>
    <div class="step-dots">
      <span class="dot active" data-step="1"></span>
      <span class="dot" data-step="2"></span>
      <span class="dot" data-step="3"></span>
      <span class="dot" data-step="4"></span>
    </div>
    <button class="step-btn" id="nextBtn">→</button>
  </div>

  <script>
    const btnSeq = document.getElementById('btnSequence');
    const btnSer = document.getElementById('btnSeries');
    const termsRow = document.getElementById('termsRow');
    const seps = [document.getElementById('sep1'), document.getElementById('sep2'), document.getElementById('sep3')];
    const boxes = [document.getElementById('box1'), document.getElementById('box2'), document.getElementById('box3'), document.getElementById('box4')];
    const segs = [document.getElementById('seg1'), document.getElementById('seg2'), document.getElementById('seg3'), document.getElementById('seg4')];
    const sumLabel = document.getElementById('sumLabel');
    const sumDisplay = document.getElementById('sumDisplay');
    const stepper = document.getElementById('stepperControls');
    const prevBtn = document.getElementById('prevBtn');
    const nextBtn = document.getElementById('nextBtn');
    const dots = document.querySelectorAll('.dot');

    const sums = [2, 8, 26, 80];
    const terms = [2, 6, 18, 54];
    let mode = 'sequence';
    let currentStep = 4;

    function renderSequence() {
      mode = 'sequence';
      btnSeq.classList.add('active');
      btnSer.classList.remove('active');
      termsRow.classList.add('show-mult');
      stepper.style.visibility = 'hidden';

      seps.forEach(s => s.textContent = ',');
      boxes.forEach(b => b.classList.add('active'));
      segs.forEach(s => s.classList.add('lit'));

      sumLabel.textContent = 'Common Ratio (r)';
      sumDisplay.textContent = '× 3';
    }

    function renderSeries(step) {
      mode = 'series';
      currentStep = step;
      btnSeq.classList.remove('active');
      btnSer.classList.add('active');
      termsRow.classList.remove('show-mult');
      stepper.style.visibility = 'visible';

      seps.forEach(s => s.textContent = '+');

      boxes.forEach((b, i) => {
        if (i < step) {
          b.classList.add('active');
        } else {
          b.classList.remove('active');
        }
      });

      segs.forEach((s, i) => {
        if (i < step) {
          s.classList.add('lit');
        } else {
          s.classList.remove('lit');
        }
      });

      dots.forEach((d, i) => {
        d.classList.toggle('active', i === step - 1);
      });

      prevBtn.disabled = step === 1;
      nextBtn.disabled = step === 4;

      sumLabel.textContent = `Sum S${step}`;
      sumDisplay.textContent = `= ${sums[step - 1]}`;
    }

    btnSeq.addEventListener('click', renderSequence);
    btnSer.addEventListener('click', () => renderSeries(currentStep));

    prevBtn.addEventListener('click', () => {
      if (currentStep > 1) renderSeries(currentStep - 1);
    });

    nextBtn.addEventListener('click', () => {
      if (currentStep < 4) renderSeries(currentStep + 1);
    });

    dots.forEach((dot, index) => {
      dot.addEventListener('click', () => renderSeries(index + 1));
    });

    boxes.forEach((box, index) => {
      box.addEventListener('click', () => {
        if (mode === 'series') {
          renderSeries(index + 1);
        }
      });
    });
  </script>
</body>
</html>

How do we track the running total as we add terms one by one?

A **partial sum**, denoted $S_n$, is the cumulative total of the first $n$ terms of a series.

A **geometric series** is the indicated sum of the terms of a geometric sequence: $S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1} = \sum_{k=1}^n a_1 r^{k-1}$.

In this sigma notation, the index starts at $k=1$ to produce the initial term $a_1$ and ends at $n$ for the $n$-th term.

For the series $2 + 6 + 18 + 54$, the first partial sum is $S_1 = 2$, the second is $S_2 = 2 + 6 = 8$, the third is $S_3 = 2 + 6 + 18 = 26$, and the fourth is $S_4 = 80$.

In general, a **geometric series** is the indicated sum of the terms of a geometric sequence: $S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1} = \sum_{k=1}^n a_1 r^{k-1}$.

Using **sigma notation**, the summation index starts at $k=1$ with the first term $a_1 r^0 = a_1$ and ends at $k=n$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Geometric Series Partial Sums</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css">
  <script src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .nav-tabs {
      display: flex;
      gap: 6px;
      margin-bottom: 8px;
      width: 100%;
      max-width: 420px;
    }

    .tab-btn {
      flex: 1;
      padding: 8px 4px;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      font-size: 14px;
      font-weight: 600;
      border-radius: 8px;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 4px rgba(2, 132, 199, 0.25);
    }

    /* Main visual stage */
    .stage {
      display: flex;
      flex-direction: column;
      align-items: center;
      width: 100%;
      max-width: 440px;
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 14px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06);
      border: 1px solid #e6e6e6;
    }

    /* Term sequence progression */
    .terms-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      margin-bottom: 12px;
      padding-bottom: 10px;
      border-bottom: 1px solid #f1f5f9;
    }

    .term-item {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 3px;
      opacity: 0.35;
      transition: opacity 0.3s ease;
    }

    .term-item.active {
      opacity: 1;
    }

    .term-badge {
      font-size: 11px;
      color: #64748b;
      font-weight: 600;
    }

    .term-val {
      font-size: 15px;
      font-weight: 700;
      padding: 4px 8px;
      border-radius: 6px;
      background: #f1f5f9;
      color: #334155;
    }

    .term-item.active .term-val {
      background: #e0f2fe;
      color: #0369a1;
      box-shadow: 0 0 0 1.5px #0284c7;
    }

    .multiplier {
      font-size: 11px;
      color: #0284c7;
      font-weight: 700;
    }

    /* Stacked visual representation */
    .stack-container {
      display: flex;
      align-items: flex-end;
      justify-content: center;
      gap: 12px;
      height: 140px;
      width: 100%;
      margin: 8px 0;
      padding: 0 8px;
    }

    .stack-col {
      flex: 1;
      display: flex;
      flex-direction: column;
      align-items: center;
      height: 100%;
      justify-content: flex-end;
    }

    .bar-group {
      width: 100%;
      max-width: 48px;
      display: flex;
      flex-direction: column-reverse;
      background: #f8fafc;
      border-radius: 6px;
      border: 1px dashed #cbd5e1;
      overflow: hidden;
      transition: all 0.3s ease;
      height: 100px;
      justify-content: flex-start;
    }

    .stack-col.active .bar-group {
      border: 1px solid #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.15);
    }

    .bar-segment {
      width: 100%;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 10px;
      font-weight: 700;
      color: #ffffff;
      transition: height 0.3s ease;
    }

    .c-t1 { background: #38bdf8; }
    .c-t2 { background: #0284c7; }
    .c-t3 { background: #0369a1; }
    .c-t4 { background: #0c4a6e; }

    .stack-label {
      font-size: 12px;
      font-weight: 700;
      margin-top: 6px;
      color: #64748b;
    }

    .stack-col.active .stack-label {
      color: #0284c7;
    }

    /* Equation / Sum Breakdown */
    .breakdown-card {
      width: 100%;
      background: #f8fafc;
      border-radius: 8px;
      padding: 8px 12px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      border: 1px solid #e2e8f0;
      min-height: 44px;
    }

    .math-eq {
      font-size: 15px;
      color: #0f172a;
    }

    .controls {
      display: flex;
      align-items: center;
      gap: 12px;
      margin-top: 10px;
    }

    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .ctrl-btn:hover {
      background: #f1f5f9;
    }
  </style>
</head>
<body>

  
  <div class="nav-tabs">
    <button class="tab-btn active" onclick="setStep(0)">S₁</button>
    <button class="tab-btn" onclick="setStep(1)">S₂</button>
    <button class="tab-btn" onclick="setStep(2)">S₃</button>
    <button class="tab-btn" onclick="setStep(3)">S₄</button>
  </div>

  
  <div class="stage">
    
    <div class="terms-row">
      <div class="term-item active" id="term-0">
        <span class="term-badge">a₁</span>
        <span class="term-val">2</span>
      </div>
      <span class="multiplier">×3 →</span>
      <div class="term-item" id="term-1">
        <span class="term-badge">a₂</span>
        <span class="term-val">6</span>
      </div>
      <span class="multiplier">×3 →</span>
      <div class="term-item" id="term-2">
        <span class="term-badge">a₃</span>
        <span class="term-val">18</span>
      </div>
      <span class="multiplier">×3 →</span>
      <div class="term-item" id="term-3">
        <span class="term-badge">a₄</span>
        <span class="term-val">54</span>
      </div>
    </div>

    
    <div class="stack-container">
      <div class="stack-col active" id="col-0" onclick="setStep(0)">
        <div class="bar-group">
          <div class="bar-segment c-t1" style="height: 3px;">2</div>
        </div>
        <div class="stack-label">S₁ = 2</div>
      </div>

      <div class="stack-col" id="col-1" onclick="setStep(1)">
        <div class="bar-group">
          <div class="bar-segment c-t1" style="height: 3px;"></div>
          <div class="bar-segment c-t2" style="height: 7px;">6</div>
        </div>
        <div class="stack-label">S₂ = 8</div>
      </div>

      <div class="stack-col" id="col-2" onclick="setStep(2)">
        <div class="bar-group">
          <div class="bar-segment c-t1" style="height: 3px;"></div>
          <div class="bar-segment c-t2" style="height: 7px;"></div>
          <div class="bar-segment c-t3" style="height: 22px;">18</div>
        </div>
        <div class="stack-label">S₃ = 26</div>
      </div>

      <div class="stack-col" id="col-3" onclick="setStep(3)">
        <div class="bar-group">
          <div class="bar-segment c-t1" style="height: 3px;"></div>
          <div class="bar-segment c-t2" style="height: 7px;"></div>
          <div class="bar-segment c-t3" style="height: 22px;"></div>
          <div class="bar-segment c-t4" style="height: 68px;">54</div>
        </div>
        <div class="stack-label">S₄ = 80</div>
      </div>
    </div>

    
    <div class="breakdown-card">
      <div id="math-render" class="math-eq"></div>
    </div>
  </div>

  
  <div class="controls">
    <button class="ctrl-btn" id="playBtn" onclick="togglePlay()">
      <span id="playIcon">⏸</span> Pause
    </button>
  </div>

  <script>
    const equations = [
      "S_1 = 2",
      "S_2 = S_1 + 6 = 8",
      "S_3 = S_2 + 18 = 26",
      "S_4 = S_3 + 54 = 80"
    ];

    let current = 0;
    let playing = true;
    let timer = null;

    function renderMath(idx) {
      const el = document.getElementById('math-render');
      if (window.katex) {
        katex.render(equations[idx], el, { throwOnError: false });
      } else {
        el.textContent = equations[idx];
      }
    }

    function setStep(idx) {
      current = idx;
      
      // Update tabs
      const tabs = document.querySelectorAll('.tab-btn');
      tabs.forEach((t, i) => t.classList.toggle('active', i === idx));

      // Update term highlights
      for (let i = 0; i < 4; i++) {
        const term = document.getElementById('term-' + i);
        if (term) term.classList.toggle('active', i <= idx);
        
        const col = document.getElementById('col-' + i);
        if (col) col.classList.toggle('active', i === idx);
      }

      renderMath(idx);
    }

    function nextStep() {
      current = (current + 1) % 4;
      setStep(current);
    }

    function startTimer() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3200);
    }

    function togglePlay() {
      playing = !playing;
      const icon = document.getElementById('playIcon');
      const btn = document.getElementById('playBtn');
      if (playing) {
        icon.textContent = '⏸';
        btn.childNodes[2].textContent = ' Pause';
        startTimer();
      } else {
        icon.textContent = '▶';
        btn.childNodes[2].textContent = ' Play';
        clearInterval(timer);
      }
    }

    // Init
    window.addEventListener('DOMContentLoaded', () => {
      setStep(0);
      startTimer();
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</body>
</html>

What if you are only given the sequence of partial sums $S_n$, and the original terms are hidden?

The first term $a_1$ is always identical to the first partial sum: $a_1 = S_1$.

To find any subsequent term $a_n$, subtract the previous cumulative total from the current total: $a_n = S_n - S_{n-1}$.

Once you have the individual terms $a_1, a_2, a_3, \dots$, find the common ratio $r$ by dividing any term by its predecessor: $r = \frac{a_n}{a_{n-1}}$.

## Worked Example: Decoding a Mystery Series

Suppose a series has partial sums given by $S_1 = 5$, $S_2 = 15$, and $S_3 = 35$. Let's determine the first term $a_1$ and the common ratio $r$.

First, extract each individual term using subtraction: $a_1 = S_1 = 5$, $a_2 = S_2 - S_1 = 15 - 5 = 10$, and $a_3 = S_3 - S_2 = 35 - 15 = 20$.

Next, compute the ratios between consecutive terms: $\frac{a_2}{a_1} = \frac{10}{5} = 2$ and $\frac{a_3}{a_2} = \frac{20}{10} = 2$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Geometric Series Visualization</title>
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      max-width: 480px;
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      align-items: flex-end;
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    .bar-col {
      flex: 1;
      max-width: 68px;
      height: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
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    .bar-track {
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    .bar {
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      align-items: flex-start;
      padding-top: 6px;
      opacity: 0.35;
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    .bar.active {
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      opacity: 0;
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    .bar.active .val-label {
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    .term-label {
      margin-top: 8px;
      font-size: 14px;
      font-weight: 600;
      color: #94a3b8;
      transition: color 0.3s ease;
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    .term-label.active {
      color: #0284c7;
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    .ratio-badge {
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      color: #0284c7;
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      padding: 3px 8px;
      border-radius: 12px;
      font-size: 12px;
      font-weight: 700;
      opacity: 0;
      transform: translateY(4px);
      transition: opacity 0.3s ease, transform 0.3s ease;
      z-index: 5;
    }

    .ratio-badge.r1 { left: calc(25% - 20px); }
    .ratio-badge.r2 { left: calc(50% - 20px); }
    .ratio-badge.r3 { left: calc(75% - 20px); }

    .ratio-badge.visible {
      opacity: 1;
      transform: translateY(0);
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    .summary-card {
      width: 100%;
      max-width: 480px;
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 16px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06), 0 1px 2px rgba(0,0,0,0.04);
      display: flex;
      align-items: center;
      justify-content: space-between;
      border: 1px solid #f1f5f9;
    }

    .sum-formula {
      font-size: 15px;
      color: #475569;
      font-weight: 500;
      letter-spacing: 0.2px;
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    .sum-total {
      font-size: 18px;
      font-weight: 800;
      color: #0284c7;
    }

    .play-btn {
      width: 32px;
      height: 32px;
      border-radius: 50%;
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      justify-content: center;
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      background: #e2e8f0;
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  </style>
</head>
<body>

  <div class="controls">
    <button class="btn active" onclick="setStep(0)">n = 1</button>
    <button class="btn" onclick="setStep(1)">n = 2</button>
    <button class="btn" onclick="setStep(2)">n = 3</button>
    <button class="btn" onclick="setStep(3)">n = 4</button>
    <button class="play-btn" id="playBtn" onclick="togglePlay()" title="Auto play">▶</button>
  </div>

  <div class="stage">
    
    <div class="ratio-badge r1" id="r1">× 2</div>
    <div class="ratio-badge r2" id="r2">× 2</div>
    <div class="ratio-badge r3" id="r3">× 2</div>

    
    <div class="bar-col">
      <div class="bar-track">
        <div class="bar active" id="bar0" style="height: 20px;">
          <span class="val-label">5</span>
        </div>
      </div>
      <div class="term-label active" id="lbl0">a₁</div>
    </div>

    
    <div class="bar-col">
      <div class="bar-track">
        <div class="bar" id="bar1" style="height: 40px;">
          <span class="val-label">10</span>
        </div>
      </div>
      <div class="term-label" id="lbl1">a₂</div>
    </div>

    
    <div class="bar-col">
      <div class="bar-track">
        <div class="bar" id="bar2" style="height: 80px;">
          <span class="val-label">20</span>
        </div>
      </div>
      <div class="term-label" id="lbl2">a₃</div>
    </div>

    
    <div class="bar-col">
      <div class="bar-track">
        <div class="bar" id="bar3" style="height: 160px;">
          <span class="val-label">40</span>
        </div>
      </div>
      <div class="term-label" id="lbl3">a₄</div>
    </div>
  </div>

  <div class="summary-card">
    <div class="sum-formula" id="formulaText">S₁ = 5</div>
    <div class="sum-total" id="sumText">Total: 5</div>
  </div>

  <script>
    const data = [
      { val: 5, sum: 5, formula: "S₁ = 5" },
      { val: 10, sum: 15, formula: "S₂ = 5 + 10" },
      { val: 20, sum: 35, formula: "S₃ = 5 + 10 + 20" },
      { val: 40, sum: 75, formula: "S₄ = 5 + 10 + 20 + 40" }
    ];

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      // Multipliers
      document.getElementById('r1').classList.toggle('visible', currentStep >= 1);
      document.getElementById('r2').classList.toggle('visible', currentStep >= 2);
      document.getElementById('r3').classList.toggle('visible', currentStep >= 3);

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      document.getElementById('sumText').textContent = 'Total: ' + data[currentStep].sum;
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          updateView();
        }, 1800);
      }
    }

    updateView();
  </script>
</body>
</html>

In the 3rd century BCE, Archimedes used partial sums of geometric series to calculate the exact area bound by a parabola, marking one of the earliest recorded uses of series in mathematical history.

## Pitfall: Dividing Partial Sums Directly

A common mistake is dividing the partial sums directly, such as calculating $\frac{S_2}{S_1} = \frac{15}{5} = 3$ and assuming $r = 3$.

Always subtract partial sums first to isolate the actual terms ($a_n$) before calculating the ratio, because $r$ describes how individual terms scale, not the cumulative totals.

math

Understanding Geometric Series. Define a geometric series and identify the first term and common ratio from a sequence of partial sums.

Geometric Series Introduction & Finite Geometric Series Formula

Algebra 2, Unit 2, Lesson 25: How Much Did You Add? (Finite Geometric Series Sums)

Algebra 2 Skill: Find the Sum of a Finite Geometric Series

To lead 9th-10th grade students to global mastery of this standard (HSA.SSE.B.4), structure the lesson around the concept of 'decoding cumulative data.' Begin with an intuitive scenario (e.g., total daily views of a viral post) where students are given only cumulative totals ($S_n$) and must infer daily amounts ($a_n$). Emphasize the algebraic bridge $a_n = S_n - S_{n-1}$ before formalizing the common ratio formula $r = \frac{S_2 - S_1}{S_1}$. Reinforce algebraic structure (MP7) when deriving the sum formula $S_n(1-r) = a_1(1-r^n)$ to ensure students understand the formula as an efficient calculation shortcut rather than an isolated fact to memorize.

Understanding Geometric Series

Sequences vs. Series

Partial Sums