{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r03-infinite-series-0027",
  "pilot_number": "R0034-17702",
  "number": 17702,
  "family_id": "qf-mod34-series-geometric-sum",
  "title": {
    "zh-Hant": "第 17702 題：無窮級數",
    "en": "Question 17702: The Geometric-Series Formula Retains Its Absolute-Ratio Premise"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r03-infinite-series-0027",
    "batch_id": "batch-20261014-advance-20261103-r03-infinite-series-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-infinite-series"
    ],
    "title_zh": "無窮級數",
    "stem_zh": "固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[2,3],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)",
      "explanation_zh": "Wave5第三層核對=(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "無窮級數"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-series-geometric-sum",
      "seed": 202611033027,
      "parameters": {
        "a": [
          2,
          3
        ],
        "r": [
          1,
          2
        ],
        "domain": "Exact_symbolic_series_certificate",
        "operation": "series-geometric-sum",
        "semantic_template": "qf-mod34-series-geometric-sum/{\"a\":[2,3],\"r\":[1,2]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r03-infinite-series-0027",
    "batch_id": "batch-20261014-advance-20261103-r03-infinite-series-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1a6d3cf2672a0d72392f6f9cb445147473a622fd427fe1f226e9a09d77073b90",
      "family_id": "qf-mod34-series-geometric-sum",
      "seed": 202611033027,
      "semantic_template": "qf-mod34-series-geometric-sum/{\"a\":[2,3],\"r\":[1,2]}"
    },
    "fields": {
      "title": "The Geometric-Series Formula Retains Its Absolute-Ratio Premise",
      "stem": "Use the infinite series with term a*r^k for k=0,1,..., exact a=2/3 and r=1/2, requiring |r|<1. Give (a,r,infinite_sum,absolute_ratio_below_one_symbolic_tail).",
      "answer": {
        "kind": "expression",
        "value": "(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)",
        "explanation": "Under the stated absolute-ratio premise calculate a/(1−r) exactly and use the explicit geometric tail certificate. Inputs outside this certificate domain are rejected; passing finitely many partial sums is not the proof. The exact result, retaining the requested order, is (2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)."
      },
      "skills": [
        "The Geometric-Series Formula Retains Its Absolute-Ratio Premise"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[2,3],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。",
        "en": "Use the infinite series with term a*r^k for k=0,1,..., exact a=2/3 and r=1/2, requiring |r|<1. Give (a,r,infinite_sum,absolute_ratio_below_one_symbolic_tail)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "signed geometric series one minus effective ratio := sub(1, 1/2)",
      "after": "signed geometric series one minus effective ratio = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 signed geometric series one minus effective ratio，不以小數近似替代。",
          "en": "Recompute signed geometric series one minus effective ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "signed geometric series one minus effective ratio"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "signed geometric series exact infinite candidate sum := div(2/3, 1/2)",
      "after": "signed geometric series exact infinite candidate sum = 4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 signed geometric series exact infinite candidate sum，不以小數近似替代。",
          "en": "Recompute signed geometric series exact infinite candidate sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2/3",
          "1/2"
        ],
        "output": "4/3",
        "label": "signed geometric series exact infinite candidate sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "signed geometric series:original SIGNED a=2/3;SIGNED r=1/2;absolute ratio=1/2<1;effective terms=(2/3)*(1/2)^k,k=0,1,...",
      "after": "candidate sum=4/3;effective denominator1-ratio=1/2 is nonzero",
      "relation": "implies",
      "rule_id": "series_geometric_exact_input_roles_and_absolute_ratio_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先保留原a、r的符號，依本題是否求絕對值級數才換有效係數及ratio；|r|<1是這份證書域，不用有限樣本或另種零係數例外改寫題目。",
        "en": "Retain signed a and r.Change to effective absolute values only when this operation asks for the absolute series.The |r|<1 guard is this certificate's domain;finite samples or an unrelated zero-coefficient exception do not change it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "signed geometric series:for ANY N>=1,the first N terms start at k0 with leading=2/3,ratio=1/2",
      "after": "(1-ratio)*S_N=(2/3)*(1-(1/2)^N);S_N=(2/3)*(1-(1/2)^N)/(1/2);candidate minus S_N=(2/3)*(1/2)^N/(1/2)",
      "relation": "implies",
      "rule_id": "series_geometric_all_N_finite_cancellation_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把有限和乘(1−ratio)，相鄰冪次抵消後只留首尾；這個恆等式對任意N≥1，並不是看幾個partial sums猜公式。",
        "en": "Multiply the finite sum by1−ratio and cancel adjacent powers,leaving endpoints.This identity holds for arbitrary N≥1,not a formula guessed from several partial sums."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "signed geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(2/3)*abs(1/2)^N/abs(1/2)",
      "after": "geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate",
      "relation": "implies",
      "rule_id": "series_geometric_remainder_limit_not_finite_prefix_inference",
      "conditions": [],
      "explanation": {
        "zh-Hant": "嚴格小於1的幾何冪趨零，固定有限係數與非零分母保餘項趨零，因此partial-sum sequence才有此極限。",
        "en": "Powers of a ratio with absolute value strictly below1 tend to0.Fixed finite coefficients and a nonzero denominator preserve this remainder limit,which proves convergence of the partial-sum sequence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "2/3",
      "1/2",
      "4/3",
      "absolute_ratio_below_one_symbolic_tail"
    ]
  },
  "conclusion": {
    "canonical_value": "(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
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      "batch_id": "batch-20261014-advance-20261103-r03-infinite-series-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r03-infinite-series-foundations/questions.jsonl",
      "batch_sha256": "312c02ec0cfac82c5f3d25ced63c9d3a9b9b58bd4a3fae40456906f9b2dab03c",
      "original_line": 27,
      "original_line_sha256": "53b83aa422427d4b55d77e3183770383f4763dc038052637d8041202bb7689c5",
      "question_compact_sha256": "1a6d3cf2672a0d72392f6f9cb445147473a622fd427fe1f226e9a09d77073b90",
      "english_row_compact_sha256": "055532ea769b210f3f37566c57ea937ef8954043195b777f1af10b6a5f13f14b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r03-infinite-series-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r03-infinite-series-foundations.en.jsonl"
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    "mko_bindings": {
      "mko-infinite-series": {
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        "original_path": "public/data/mko/infinite-series.json",
        "english_path": "public/data/i18n/editions/mko-infinite-series.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2",
    "original_candidate_sha256": "0ceb6ecf3d3d702966318ec6cece4014f51e7d6a9ea6e953d458c3868148f8d0",
    "original_candidate_line_sha256": "0ceb6ecf3d3d702966318ec6cece4014f51e7d6a9ea6e953d458c3868148f8d0"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p35",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336",
      "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "80d381d8199cca5ae0e4d5442fe95171312e3a91c9c8466992e9c70c2bcee6bc",
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    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}