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[
{
"slide": 1,
"fragments": [
{
"fragment_index": -1,
"text_description": "Geometric Series\nPatterns that multiply, formulas that amplify.",
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]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": 1,
"text_description": "Key Idea",
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},
{
"fragment_index": 2,
"text_description": "Geometric Sequence",
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},
{
"fragment_index": 3,
"text_description": "A geometric sequence multiplies each term by a constant ratio \\(r\\). First term: \\(a_1\\). General term: \\(a_n = a_1 r^{\\,n-1}\\).",
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},
{
"fragment_index": 4,
"text_description": "Identify \\(a_1\\) and find \\(r = \\frac{a_2}{a_1}\\) to describe any geometric sequence.",
"image_description": ""
}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Find the Ratio r\nGoal: practise extracting the common ratio from raw sequence data.",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\nSpot Consecutive Terms\nFrom the sequence \\(3, 6, 12, 24, \\dots\\) pick pairs: \\(3,6\\); \\(6,12\\); \\(12,24\\).",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\nDivide to Form Fractions\nCompute \\( \\frac{6}{3}, \\frac{12}{6}, \\frac{24}{12} \\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\nSimplify ⇒ Common Ratio\nEach fraction simplifies to 2, giving the common ratio \\( r = 2 \\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Pro Tip:\nIf every consecutive pair gives the same quotient, the sequence is geometric.",
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}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": -1,
"text_description": "General Term\nApplications",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "\\[a_n = a_1\\, r^{(n-1)}\\]\nThe nth-term formula lets you jump straight to any term in the sequence.\nState it clearly and interpret each variable correctly.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Variable Definitions\n\\(a_1\\)\nfirst term\n\\(r\\)\ncommon ratio\n\\(n\\)\nterm position",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Calculate distant terms\nFind the 50th or 100th term quickly.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Model exponential growth\nDescribe interest, population or radioactive decay.",
"image_description": ""
}
]
},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nCorrect!\nCorrect! \\(5 \\times 81 = 405\\).\nIncorrect\nRevisit the exponent: \\(n - 1 = 4\\).\nconst correctOption = 1;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('slide-05-x7b9q2-submitBtn');\n const feedbackCorrect = document.getElementById('slide-05-x7b9q2-feedback-correct');\n const feedbackIncorrect = document.getElementById('slide-05-x7b9q2-feedback-incorrect');\n\n let selectedOption = null;\n\n answerCards.forEach((card, index) => {\n card.addEventListener('click', () => {\n answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n card.classList.add('border-blue-500', 'bg-blue-50');\n selectedOption = index;\n });\n });\n\n submitBtn.addEventListener('click', () => {\n if (selectedOption === null) return;\n\n if (selectedOption === correctOption) {\n feedbackCorrect.classList.remove('hidden');\n feedbackIncorrect.classList.add('hidden');\n } else {\n feedbackIncorrect.classList.remove('hidden');\n feedbackCorrect.classList.add('hidden');\n }\n });",
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{
"fragment_index": 1,
"text_description": "Question\nFor the geometric sequence with first term \\(a_1 = 5\\) and common ratio \\(r = 3\\), what is \\(a_5\\)?",
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},
{
"fragment_index": 2,
"text_description": "1\n135\n2\n405\n3\n243\n4\n625",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Hint:\nUse \\(a_5 = 5 \\times 3^{4}\\).",
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},
{
"fragment_index": 4,
"text_description": "Submit Answer",
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}
]
},
{
"slide": 6,
"fragments": [
{
"fragment_index": -1,
"text_description": "Growth vs Decay\nCompare r = 2 and r = 0.5",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "r = 2 (blue) grows, r = 0.5 (red) decays.",
"image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/UrXq1PCVcq3iRaOTg6nXfH9FAZg3PCj9KgiwgRZR.png"
},
{
"fragment_index": 2,
"text_description": "Both sequences begin at 1 on the graph.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "With r = 2, each term doubles; the curve shoots upward, showing exponential growth.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "With r = 0.5, each term halves; the curve sinks toward the x-axis, showing exponential decay.",
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},
{
"fragment_index": 5,
"text_description": "Key Points:\nGraphical intuition: steep upward slope means growth; downward slope means decay.\nComparison: identical starts diverge quickly because r differs.\nLong-term: r > 1 → ∞, 0 < r < 1 → 0.",
"image_description": ""
}
]
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "Partial Sum \\(S_n\\)",
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},
{
"fragment_index": 1,
"text_description": "A finite geometric series sums the first \\(n\\) terms of a sequence. Know this formula by heart when \\(r \\neq 1\\).\n\\[S_n = a_1\\, \\frac{1 - r^{n}}{1 - r}\\quad\\text{for}\\; r \\ne 1\\]",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Variable Definitions\n\\(S_n\\)\nsum of first \\(n\\) terms\n\\(a_1\\)\nfirst term\n\\(r\\)\ncommon ratio\n\\(n\\)\nnumber of terms",
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},
{
"fragment_index": 3,
"text_description": "Applications\nLoan schedules\nDetermine the remaining balance after \\(n\\) equal payments.\nPopulation totals\nEstimate cumulative population after \\(n\\) years of geometric growth.",
"image_description": ""
}
]
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Where S\nn\nComes From",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "1\n\\[S_n = a_1 + a_1 r + a_1 r^2 + \\dots + a_1 r^{n-1}\\]\nWrite the series in expanded form.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "2\n\\[r S_n = a_1 r + a_1 r^2 + \\dots + a_1 r^{n}\\]\nMultiply every term by \\(r\\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "3\n\\[S_n - r S_n = a_1 - a_1 r^{n}\\]\nSubtract to eliminate the middle terms.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "4\n\\[S_n (1 - r) = a_1 (1 - r^{n})\\]\nFactor \\(S_n\\) on the left.",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "5\n\\[S_n = a_1 \\frac{1 - r^{n}}{1 - r}\\]\nDivide by \\(1 - r\\) to isolate \\(S_n\\).",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Key Insight:\nTelescoping quickly collapses the series.",
"image_description": ""
}
]
},
{
"slide": 9,
"fragments": []
},
{
"slide": 10,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "🔑\nMultiply, don’t add\nEach step scales by \\(r\\); think multiplication, not addition.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "📍\na\nn\nformula\nJump to term \\(a_n = a_1 r^{n-1}\\) instantly.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "🧮\nSum S\nn\nFinite sum: \\(S_n = a_1 \\frac{1 - r^{n}}{1 - r}\\).",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "♾️\nConvergence rule\nInfinite series converges only when \\(|r| < 1\\).",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "🌍\nReal-world power\nGoverns compound interest, population change and exponential decay.",
"image_description": ""
}
]
}
]