{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0001","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac23fe23fe3ad361dbe5e7d2ca30a624666874ec40b4d227549a5f02cec7e60","family_id":"qf-sequence-explicit-term","seed":202609305001,"semantic_template":"qf-sequence-explicit-term/{\"d\":-3,\"n\":0,\"operation\":\"explicit_term\",\"u\":-12}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-12)+(-3)n for n∈N={0,1,2,…}. Find the term at n=0.","answer":{"kind":"short_text","value":"-12","explanation":"Substitute the explicit zero-based index: a_0=-12+(-3)×0=-12. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0002","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc35641ea89534350f4799e7ca23d63e45eec2374b8d00351a167d54c8578f6a","family_id":"qf-sequence-explicit-term","seed":202609305002,"semantic_template":"qf-sequence-explicit-term/{\"d\":-2,\"n\":1,\"operation\":\"explicit_term\",\"u\":-11}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-11)+(-2)n for n∈N={0,1,2,…}. Find the term at n=1.","answer":{"kind":"short_text","value":"-13","explanation":"Substitute the explicit zero-based index: a_1=-11+(-2)×1=-13. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0003","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bacbf6bdac70b577b6f9abbc4cc0d19ec5de49f8ea4d8e1063da70fb16dec961","family_id":"qf-sequence-explicit-term","seed":202609305003,"semantic_template":"qf-sequence-explicit-term/{\"d\":-1,\"n\":2,\"operation\":\"explicit_term\",\"u\":-10}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-10)+(-1)n for n∈N={0,1,2,…}. Find the term at n=2.","answer":{"kind":"short_text","value":"-12","explanation":"Substitute the explicit zero-based index: a_2=-10+(-1)×2=-12. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0004","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6eb74eda845953cd514ba6e6b73a0c678761c8c889eaf4788905ec55eb6b83c","family_id":"qf-sequence-explicit-term","seed":202609305004,"semantic_template":"qf-sequence-explicit-term/{\"d\":0,\"n\":3,\"operation\":\"explicit_term\",\"u\":-9}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-9)+(0)n for n∈N={0,1,2,…}. Find the term at n=3.","answer":{"kind":"short_text","value":"-9","explanation":"Substitute the explicit zero-based index: a_3=-9+(0)×3=-9. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0005","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ce643b00a96509cd2a912a456d836d14f7eb25a480e005cf3aa0735a2b4311b","family_id":"qf-sequence-explicit-term","seed":202609305005,"semantic_template":"qf-sequence-explicit-term/{\"d\":1,\"n\":4,\"operation\":\"explicit_term\",\"u\":-8}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-8)+(1)n for n∈N={0,1,2,…}. Find the term at n=4.","answer":{"kind":"short_text","value":"-4","explanation":"Substitute the explicit zero-based index: a_4=-8+(1)×4=-4. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0006","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dc592d32043cc6a9cc3a5c4954f812e7de343dc8736ffd77f48c7626d8459cd","family_id":"qf-sequence-explicit-term","seed":202609305006,"semantic_template":"qf-sequence-explicit-term/{\"d\":2,\"n\":5,\"operation\":\"explicit_term\",\"u\":-7}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-7)+(2)n for n∈N={0,1,2,…}. Find the term at n=5.","answer":{"kind":"short_text","value":"3","explanation":"Substitute the explicit zero-based index: a_5=-7+(2)×5=3. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0007","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6758008fe9d5450d3008f09805b41ad582d757a9e047537014ddaf9748a16769","family_id":"qf-sequence-explicit-term","seed":202609305007,"semantic_template":"qf-sequence-explicit-term/{\"d\":3,\"n\":6,\"operation\":\"explicit_term\",\"u\":-6}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-6)+(3)n for n∈N={0,1,2,…}. Find the term at n=6.","answer":{"kind":"short_text","value":"12","explanation":"Substitute the explicit zero-based index: a_6=-6+(3)×6=12. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0008","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfd2964b53643890c03512215b878c0d0d5fa0908a55c773dfe00858f553a5ce","family_id":"qf-sequence-explicit-term","seed":202609305008,"semantic_template":"qf-sequence-explicit-term/{\"d\":-3,\"n\":7,\"operation\":\"explicit_term\",\"u\":-5}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-5)+(-3)n for n∈N={0,1,2,…}. Find the term at n=7.","answer":{"kind":"short_text","value":"-26","explanation":"Substitute the explicit zero-based index: a_7=-5+(-3)×7=-26. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0009","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"602db05e172d4b8df29085c7ca6ac3c139695138bfe4b2ac1264898a2346d08d","family_id":"qf-sequence-explicit-term","seed":202609305009,"semantic_template":"qf-sequence-explicit-term/{\"d\":-2,\"n\":8,\"operation\":\"explicit_term\",\"u\":-4}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-4)+(-2)n for n∈N={0,1,2,…}. Find the term at n=8.","answer":{"kind":"short_text","value":"-20","explanation":"Substitute the explicit zero-based index: a_8=-4+(-2)×8=-20. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0010","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa093e187ca44166980960edf4ba30d425041567d5098b9087a3c68e780d76a1","family_id":"qf-sequence-explicit-term","seed":202609305010,"semantic_template":"qf-sequence-explicit-term/{\"d\":-1,\"n\":9,\"operation\":\"explicit_term\",\"u\":-3}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-3)+(-1)n for n∈N={0,1,2,…}. Find the term at n=9.","answer":{"kind":"short_text","value":"-12","explanation":"Substitute the explicit zero-based index: a_9=-3+(-1)×9=-12. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0011","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb8412480dac2771d5fbe83085d3edd77162a9ae7200f9ea457de9d72e510ecf","family_id":"qf-sequence-explicit-term","seed":202609305011,"semantic_template":"qf-sequence-explicit-term/{\"d\":0,\"n\":0,\"operation\":\"explicit_term\",\"u\":-2}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-2)+(0)n for n∈N={0,1,2,…}. Find the term at n=0.","answer":{"kind":"short_text","value":"-2","explanation":"Substitute the explicit zero-based index: a_0=-2+(0)×0=-2. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0012","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cc24ec1bbb3dca845ee828e0c796e06d1e612c5b0aee557f54ba286999084d1","family_id":"qf-sequence-explicit-term","seed":202609305012,"semantic_template":"qf-sequence-explicit-term/{\"d\":1,\"n\":1,\"operation\":\"explicit_term\",\"u\":-1}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(-1)+(1)n for n∈N={0,1,2,…}. Find the term at n=1.","answer":{"kind":"short_text","value":"0","explanation":"Substitute the explicit zero-based index: a_1=-1+(1)×1=0. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0013","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81e50ed39a7d47a303b396f31496300679282a34e6c3882bb6d45a8c208f9801","family_id":"qf-sequence-explicit-term","seed":202609305013,"semantic_template":"qf-sequence-explicit-term/{\"d\":2,\"n\":2,\"operation\":\"explicit_term\",\"u\":0}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(0)+(2)n for n∈N={0,1,2,…}. Find the term at n=2.","answer":{"kind":"short_text","value":"4","explanation":"Substitute the explicit zero-based index: a_2=0+(2)×2=4. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0014","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09451a47e893da6bc11c43a0a0c37577417acad3cecb1fa2d99fa8bd46e69a82","family_id":"qf-sequence-explicit-term","seed":202609305014,"semantic_template":"qf-sequence-explicit-term/{\"d\":3,\"n\":3,\"operation\":\"explicit_term\",\"u\":1}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(1)+(3)n for n∈N={0,1,2,…}. Find the term at n=3.","answer":{"kind":"short_text","value":"10","explanation":"Substitute the explicit zero-based index: a_3=1+(3)×3=10. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0015","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1cb039a3b088c8e42a4fc612843f8abbf810cf9765a291ce5efc5420d847a94","family_id":"qf-sequence-explicit-term","seed":202609305015,"semantic_template":"qf-sequence-explicit-term/{\"d\":-3,\"n\":4,\"operation\":\"explicit_term\",\"u\":2}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(2)+(-3)n for n∈N={0,1,2,…}. Find the term at n=4.","answer":{"kind":"short_text","value":"-10","explanation":"Substitute the explicit zero-based index: a_4=2+(-3)×4=-10. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0016","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068cc86fc833ae84637eeaf77f1ef67d496d15d5992daf8095dcd62e7765a127","family_id":"qf-sequence-explicit-term","seed":202609305016,"semantic_template":"qf-sequence-explicit-term/{\"d\":-2,\"n\":5,\"operation\":\"explicit_term\",\"u\":3}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(3)+(-2)n for n∈N={0,1,2,…}. Find the term at n=5.","answer":{"kind":"short_text","value":"-7","explanation":"Substitute the explicit zero-based index: a_5=3+(-2)×5=-7. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0017","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac293f1d6b6aba984eee1c79c54f97efcffe872ebdae07b78a5192b9ca147194","family_id":"qf-sequence-explicit-term","seed":202609305017,"semantic_template":"qf-sequence-explicit-term/{\"d\":-1,\"n\":6,\"operation\":\"explicit_term\",\"u\":4}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(4)+(-1)n for n∈N={0,1,2,…}. Find the term at n=6.","answer":{"kind":"short_text","value":"-2","explanation":"Substitute the explicit zero-based index: a_6=4+(-1)×6=-2. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0018","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e8aa8968e886aa743663b26c143ae7aa73373ad449ae9d659969ec9fd8960c","family_id":"qf-sequence-explicit-term","seed":202609305018,"semantic_template":"qf-sequence-explicit-term/{\"d\":0,\"n\":7,\"operation\":\"explicit_term\",\"u\":5}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(5)+(0)n for n∈N={0,1,2,…}. Find the term at n=7.","answer":{"kind":"short_text","value":"5","explanation":"Substitute the explicit zero-based index: a_7=5+(0)×7=5. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0019","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29186628dd19027e41c3db00507fdf39de57f8350620f4c79f4d3187d2be43ad","family_id":"qf-sequence-explicit-term","seed":202609305019,"semantic_template":"qf-sequence-explicit-term/{\"d\":1,\"n\":8,\"operation\":\"explicit_term\",\"u\":6}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(6)+(1)n for n∈N={0,1,2,…}. Find the term at n=8.","answer":{"kind":"short_text","value":"14","explanation":"Substitute the explicit zero-based index: a_8=6+(1)×8=14. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0020","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8f0a49f290e3444b61d6a2a0b1ffe6764d72c401b5b1d13a040d6bd961beca2","family_id":"qf-sequence-explicit-term","seed":202609305020,"semantic_template":"qf-sequence-explicit-term/{\"d\":2,\"n\":9,\"operation\":\"explicit_term\",\"u\":7}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(7)+(2)n for n∈N={0,1,2,…}. Find the term at n=9.","answer":{"kind":"short_text","value":"25","explanation":"Substitute the explicit zero-based index: a_9=7+(2)×9=25. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0021","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e046da3a04229325547182f183ee43c7cdd377d38f0006af47d126681fc1565","family_id":"qf-sequence-explicit-term","seed":202609305021,"semantic_template":"qf-sequence-explicit-term/{\"d\":3,\"n\":0,\"operation\":\"explicit_term\",\"u\":8}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(8)+(3)n for n∈N={0,1,2,…}. Find the term at n=0.","answer":{"kind":"short_text","value":"8","explanation":"Substitute the explicit zero-based index: a_0=8+(3)×0=8. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0022","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"174343740cd5efaf0d6a3230ae4934a7d0c1ee6688241d8b1de108d7ffa99df3","family_id":"qf-sequence-explicit-term","seed":202609305022,"semantic_template":"qf-sequence-explicit-term/{\"d\":-3,\"n\":1,\"operation\":\"explicit_term\",\"u\":9}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(9)+(-3)n for n∈N={0,1,2,…}. Find the term at n=1.","answer":{"kind":"short_text","value":"6","explanation":"Substitute the explicit zero-based index: a_1=9+(-3)×1=6. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0023","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c91f3b40e6546705d0f566d3d629cfe9d1a3ff826afd3af2ca9473faa629a8fb","family_id":"qf-sequence-explicit-term","seed":202609305023,"semantic_template":"qf-sequence-explicit-term/{\"d\":-2,\"n\":2,\"operation\":\"explicit_term\",\"u\":10}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(10)+(-2)n for n∈N={0,1,2,…}. Find the term at n=2.","answer":{"kind":"short_text","value":"6","explanation":"Substitute the explicit zero-based index: a_2=10+(-2)×2=6. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0024","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4470c4f39fe02c4fe586311d367ce9044629242e67a4b294046323b68accf27","family_id":"qf-sequence-explicit-term","seed":202609305024,"semantic_template":"qf-sequence-explicit-term/{\"d\":-1,\"n\":3,\"operation\":\"explicit_term\",\"u\":11}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(11)+(-1)n for n∈N={0,1,2,…}. Find the term at n=3.","answer":{"kind":"short_text","value":"8","explanation":"Substitute the explicit zero-based index: a_3=11+(-1)×3=8. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-explicit-term-0025","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"321d8c4061c3b08b77491f296e43c6a9c55c5507dc5897750347fb63cf369040","family_id":"qf-sequence-explicit-term","seed":202609305025,"semantic_template":"qf-sequence-explicit-term/{\"d\":0,\"n\":4,\"operation\":\"explicit_term\",\"u\":12}"},"fields":{"title":"Term of a Zero-Based Explicit Sequence","stem":"The real sequence is indexed from 0 and has the explicit rule a_n=(12)+(0)n for n∈N={0,1,2,…}. Find the term at n=4.","answer":{"kind":"short_text","value":"12","explanation":"Substitute the explicit zero-based index: a_4=12+(0)×4=12. No limit claim is involved."},"skills":["zero-based indexing","explicit term formula"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0001","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d81cd93c9a7de13d29129160c37ceebaf34c06dec8ea42780f64145b1add925","family_id":"qf-sequence-recursive-term","seed":202609305101,"semantic_template":"qf-sequence-recursive-term/{\"d\":-3,\"n\":0,\"operation\":\"recursive_term\",\"u\":12}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=12 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=0.","answer":{"kind":"short_text","value":"12","explanation":"Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=12."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0002","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb7b38bee070e4986a77d78cd71e057a3d2468fcc5622d61ede0e1182904eb73","family_id":"qf-sequence-recursive-term","seed":202609305102,"semantic_template":"qf-sequence-recursive-term/{\"d\":-2,\"n\":3,\"operation\":\"recursive_term\",\"u\":11}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=11 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=3.","answer":{"kind":"short_text","value":"5","explanation":"Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=5."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0003","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d75ac77d10b4e8cd4a2a5d244f08589aadfcc7dedb123a69ae71e2d96fd968b","family_id":"qf-sequence-recursive-term","seed":202609305103,"semantic_template":"qf-sequence-recursive-term/{\"d\":-1,\"n\":6,\"operation\":\"recursive_term\",\"u\":10}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=10 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=6.","answer":{"kind":"short_text","value":"4","explanation":"Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=4."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0004","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d513cda15e7f348537b4fcd8deafd6127ca9516fd72e081be4922e13425e279","family_id":"qf-sequence-recursive-term","seed":202609305104,"semantic_template":"qf-sequence-recursive-term/{\"d\":0,\"n\":9,\"operation\":\"recursive_term\",\"u\":9}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=9 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=9.","answer":{"kind":"short_text","value":"9","explanation":"Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=9."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0005","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc5dca0361db5f3d99360454da2106fa0bfccc745f8d98ea3a93d673ff386402","family_id":"qf-sequence-recursive-term","seed":202609305105,"semantic_template":"qf-sequence-recursive-term/{\"d\":1,\"n\":12,\"operation\":\"recursive_term\",\"u\":8}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=8 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=12.","answer":{"kind":"short_text","value":"20","explanation":"Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=20."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0006","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e5e36605c98efeaec771b26e9896bb7e6937869f250ed8e3206e6a060d98a16","family_id":"qf-sequence-recursive-term","seed":202609305106,"semantic_template":"qf-sequence-recursive-term/{\"d\":2,\"n\":2,\"operation\":\"recursive_term\",\"u\":7}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=7 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=2.","answer":{"kind":"short_text","value":"11","explanation":"Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=11."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0007","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c358dfd239dc3f3c2ac660cd46564820f19a645cddde3df0ffa69f79e04611a1","family_id":"qf-sequence-recursive-term","seed":202609305107,"semantic_template":"qf-sequence-recursive-term/{\"d\":3,\"n\":5,\"operation\":\"recursive_term\",\"u\":6}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=6 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=5.","answer":{"kind":"short_text","value":"21","explanation":"Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=21."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0008","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a1812fb86ae0107edb7ff86adca258dc1657541fafcfe514a5a5925f4a95a5","family_id":"qf-sequence-recursive-term","seed":202609305108,"semantic_template":"qf-sequence-recursive-term/{\"d\":-3,\"n\":8,\"operation\":\"recursive_term\",\"u\":5}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=5 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=8.","answer":{"kind":"short_text","value":"-19","explanation":"Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=-19."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0009","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"578b4b2fab224e3c6aadb7972bc6e285a4ad7e7d9f97c89caa40fe1555a0677c","family_id":"qf-sequence-recursive-term","seed":202609305109,"semantic_template":"qf-sequence-recursive-term/{\"d\":-2,\"n\":11,\"operation\":\"recursive_term\",\"u\":4}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=4 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=11.","answer":{"kind":"short_text","value":"-18","explanation":"Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-18."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0010","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fbfb199315e8be5ef91a3d0e2924221972ebd7e3cd9b63474091069aae179df","family_id":"qf-sequence-recursive-term","seed":202609305110,"semantic_template":"qf-sequence-recursive-term/{\"d\":-1,\"n\":1,\"operation\":\"recursive_term\",\"u\":3}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=3 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=1.","answer":{"kind":"short_text","value":"2","explanation":"Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=2."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0011","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"974c0b70c8b3ef9d76c52f694a7de26c1c1b35bcb5c60cf7f270bb966cb4d9f8","family_id":"qf-sequence-recursive-term","seed":202609305111,"semantic_template":"qf-sequence-recursive-term/{\"d\":0,\"n\":4,\"operation\":\"recursive_term\",\"u\":2}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=2 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=4.","answer":{"kind":"short_text","value":"2","explanation":"Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=2."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0012","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8702f79c40b2d98db5ffed5f5dad2c7fdac25e485ac8e797540f6071aaa3c98","family_id":"qf-sequence-recursive-term","seed":202609305112,"semantic_template":"qf-sequence-recursive-term/{\"d\":1,\"n\":7,\"operation\":\"recursive_term\",\"u\":1}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=1 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=7.","answer":{"kind":"short_text","value":"8","explanation":"Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=8."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0013","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7132e365105295d9c43895e3f60ebb79ec742895d75ccb8374cd09a2268a608e","family_id":"qf-sequence-recursive-term","seed":202609305113,"semantic_template":"qf-sequence-recursive-term/{\"d\":2,\"n\":10,\"operation\":\"recursive_term\",\"u\":0}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=0 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=10.","answer":{"kind":"short_text","value":"20","explanation":"Start from a_0 and perform exactly 10 recurrence update(s), obtaining a_10=20."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0014","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6129ebc8287061a45929a564b7de84b74ddc0b1e426184a5a115ac50890c659d","family_id":"qf-sequence-recursive-term","seed":202609305114,"semantic_template":"qf-sequence-recursive-term/{\"d\":3,\"n\":0,\"operation\":\"recursive_term\",\"u\":-1}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-1 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=0.","answer":{"kind":"short_text","value":"-1","explanation":"Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=-1."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0015","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3a7f3ed44b56b8aa5ea9b02dee3e259e214e24b3834b0bce3258b5f6d93078c","family_id":"qf-sequence-recursive-term","seed":202609305115,"semantic_template":"qf-sequence-recursive-term/{\"d\":-3,\"n\":3,\"operation\":\"recursive_term\",\"u\":-2}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-2 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=3.","answer":{"kind":"short_text","value":"-11","explanation":"Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=-11."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0016","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c5a7412b9f9c9764a1fb0240634eb8bd0dde4b0d582a4ebf55c731bcaa88f2","family_id":"qf-sequence-recursive-term","seed":202609305116,"semantic_template":"qf-sequence-recursive-term/{\"d\":-2,\"n\":6,\"operation\":\"recursive_term\",\"u\":-3}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-3 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=6.","answer":{"kind":"short_text","value":"-15","explanation":"Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=-15."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0017","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a76459b57523b8142cc440251242ae774a1cddc2075c9ab39d39c798d4255ce9","family_id":"qf-sequence-recursive-term","seed":202609305117,"semantic_template":"qf-sequence-recursive-term/{\"d\":-1,\"n\":9,\"operation\":\"recursive_term\",\"u\":-4}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-4 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=9.","answer":{"kind":"short_text","value":"-13","explanation":"Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=-13."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0018","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daa95d78764a073eb8dde66738b4de712f8753897848669b6448bbe4b5f33084","family_id":"qf-sequence-recursive-term","seed":202609305118,"semantic_template":"qf-sequence-recursive-term/{\"d\":0,\"n\":12,\"operation\":\"recursive_term\",\"u\":-5}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-5 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=12.","answer":{"kind":"short_text","value":"-5","explanation":"Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=-5."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0019","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13a00983b0d4f2a0d297ba0bfa2a7c0a61f1eec786131e9ecf20671a43f52214","family_id":"qf-sequence-recursive-term","seed":202609305119,"semantic_template":"qf-sequence-recursive-term/{\"d\":1,\"n\":2,\"operation\":\"recursive_term\",\"u\":-6}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-6 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=2.","answer":{"kind":"short_text","value":"-4","explanation":"Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=-4."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0020","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"991e2378cb9a5f7a406080af385dd98f262dc10d6f7373148cfa2dcd0fc154a2","family_id":"qf-sequence-recursive-term","seed":202609305120,"semantic_template":"qf-sequence-recursive-term/{\"d\":2,\"n\":5,\"operation\":\"recursive_term\",\"u\":-7}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-7 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=5.","answer":{"kind":"short_text","value":"3","explanation":"Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=3."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0021","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f28f382d07f7004f9f1a9ee26e5af72995ca41612d0d91c3924e527069701a","family_id":"qf-sequence-recursive-term","seed":202609305121,"semantic_template":"qf-sequence-recursive-term/{\"d\":3,\"n\":8,\"operation\":\"recursive_term\",\"u\":-8}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-8 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=8.","answer":{"kind":"short_text","value":"16","explanation":"Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=16."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0022","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e865693cb2a97dad812d5dc3121b4de67b4a5a15a8e89c790d7369cb67fc497e","family_id":"qf-sequence-recursive-term","seed":202609305122,"semantic_template":"qf-sequence-recursive-term/{\"d\":-3,\"n\":11,\"operation\":\"recursive_term\",\"u\":-9}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-9 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=11.","answer":{"kind":"short_text","value":"-42","explanation":"Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-42."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0023","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d68d081469bb9c92b2cf0e6e6819a8fb13ce7763b6f4bef73620b9763f3a6cad","family_id":"qf-sequence-recursive-term","seed":202609305123,"semantic_template":"qf-sequence-recursive-term/{\"d\":-2,\"n\":1,\"operation\":\"recursive_term\",\"u\":-10}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-10 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=1.","answer":{"kind":"short_text","value":"-12","explanation":"Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=-12."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0024","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05a6e4ca24fd245daff61cf3f1fd1c4afe94eb8d45df7a88f779453e679a6119","family_id":"qf-sequence-recursive-term","seed":202609305124,"semantic_template":"qf-sequence-recursive-term/{\"d\":-1,\"n\":4,\"operation\":\"recursive_term\",\"u\":-11}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-11 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=4.","answer":{"kind":"short_text","value":"-15","explanation":"Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=-15."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-recursive-term-0025","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a989dc9877b32750a0da618aed8a0824f2f2e50a17109c76841d1f6edb26889f","family_id":"qf-sequence-recursive-term","seed":202609305125,"semantic_template":"qf-sequence-recursive-term/{\"d\":0,\"n\":7,\"operation\":\"recursive_term\",\"u\":-12}"},"fields":{"title":"Term of a Recurrence with an Initial Value","stem":"The real sequence is indexed from 0 with a_0=-12 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=7.","answer":{"kind":"short_text","value":"-12","explanation":"Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=-12."},"skills":["initial conditions","finite recurrence"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0001","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62244b6dbf9f8c494f56eae9c11735c7eef3be820ba39bf05ae5e61bbf65b108","family_id":"qf-sequence-prefix","seed":202609305201,"semantic_template":"qf-sequence-prefix/{\"L\":0,\"d\":-3,\"operation\":\"prefix\",\"u\":-12}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-12)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=0; use [] when L=0.","answer":{"kind":"short_text","value":"[]","explanation":"Take exactly 0 term(s) in index order beginning at 0. The explicit-formula prefix is [], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0002","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5ef1fa6afcfc3da1da7cc8fbd8514915d9becf98a380b60767834363f15fe0","family_id":"qf-sequence-prefix","seed":202609305202,"semantic_template":"qf-sequence-prefix/{\"L\":2,\"d\":-2,\"operation\":\"prefix\",\"u\":-11}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-11)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.","answer":{"kind":"short_text","value":"[-11, -13]","explanation":"Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [-11, -13], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0003","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e1deade510125275626b2aa0223a6060295f8597fc9b7c51cc6e17748348c94","family_id":"qf-sequence-prefix","seed":202609305203,"semantic_template":"qf-sequence-prefix/{\"L\":3,\"d\":-1,\"operation\":\"prefix\",\"u\":-10}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-10)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.","answer":{"kind":"short_text","value":"[-10, -11, -12]","explanation":"Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [-10, -11, -12], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0004","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14bdb314ae91fc361d1497a6432459be605bb08511d9901a8b1f4791d58b7638","family_id":"qf-sequence-prefix","seed":202609305204,"semantic_template":"qf-sequence-prefix/{\"L\":4,\"d\":0,\"operation\":\"prefix\",\"u\":-9}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-9)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.","answer":{"kind":"short_text","value":"[-9, -9, -9, -9]","explanation":"Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [-9, -9, -9, -9], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0005","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac0e0d0d3119a9b1a6e87c3efcb81c5a4ef1824261adb0f2e75eb6c292cb82e","family_id":"qf-sequence-prefix","seed":202609305205,"semantic_template":"qf-sequence-prefix/{\"L\":5,\"d\":1,\"operation\":\"prefix\",\"u\":-8}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-8)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.","answer":{"kind":"short_text","value":"[-8, -7, -6, -5, -4]","explanation":"Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [-8, -7, -6, -5, -4], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0006","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af2b7e293cb6076c85f987dbdb49ade7b2dcb06993032e2ca654a6df0801e1f","family_id":"qf-sequence-prefix","seed":202609305206,"semantic_template":"qf-sequence-prefix/{\"L\":6,\"d\":2,\"operation\":\"prefix\",\"u\":-7}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.","answer":{"kind":"short_text","value":"[-7, -5, -3, -1, 1, 3]","explanation":"Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [-7, -5, -3, -1, 1, 3], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0007","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7052846b1589ff8b144af6d24b27b3e213247d389405f0a31e452a37978786e8","family_id":"qf-sequence-prefix","seed":202609305207,"semantic_template":"qf-sequence-prefix/{\"L\":7,\"d\":3,\"operation\":\"prefix\",\"u\":-6}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-6)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.","answer":{"kind":"short_text","value":"[-6, -3, 0, 3, 6, 9, 12]","explanation":"Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [-6, -3, 0, 3, 6, 9, 12], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0008","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6e70ef56dc806f318471f9350ac13b80b5b0a81421c5cb80bdc9d990580cadf","family_id":"qf-sequence-prefix","seed":202609305208,"semantic_template":"qf-sequence-prefix/{\"L\":8,\"d\":-3,\"operation\":\"prefix\",\"u\":-5}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-5)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.","answer":{"kind":"short_text","value":"[-5, -8, -11, -14, -17, -20, -23, -26]","explanation":"Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [-5, -8, -11, -14, -17, -20, -23, -26], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0009","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"528d6a1ebe5631f5dbca3d22c87ec95833c7c56ba0ebef62a3cb9f394cf57537","family_id":"qf-sequence-prefix","seed":202609305209,"semantic_template":"qf-sequence-prefix/{\"L\":1,\"d\":-2,\"operation\":\"prefix\",\"u\":-4}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-4)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.","answer":{"kind":"short_text","value":"[-4]","explanation":"Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [-4], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0010","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b82d455c5ba15d69b50c77067ebf97c4b97728308e35ab5808dca21e00380f6","family_id":"qf-sequence-prefix","seed":202609305210,"semantic_template":"qf-sequence-prefix/{\"L\":2,\"d\":-1,\"operation\":\"prefix\",\"u\":-3}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-3)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.","answer":{"kind":"short_text","value":"[-3, -4]","explanation":"Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [-3, -4], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0011","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90af2fb66dc537e27e6cc8f6abca56946a2a7af1aacc61e54427a9cc6c007370","family_id":"qf-sequence-prefix","seed":202609305211,"semantic_template":"qf-sequence-prefix/{\"L\":3,\"d\":0,\"operation\":\"prefix\",\"u\":-2}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-2)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.","answer":{"kind":"short_text","value":"[-2, -2, -2]","explanation":"Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [-2, -2, -2], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0012","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e85554a775f8394c6a80c1253f01dc337cb9b43493c75032a1ca3ea1322bf383","family_id":"qf-sequence-prefix","seed":202609305212,"semantic_template":"qf-sequence-prefix/{\"L\":4,\"d\":1,\"operation\":\"prefix\",\"u\":-1}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(-1)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.","answer":{"kind":"short_text","value":"[-1, 0, 1, 2]","explanation":"Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [-1, 0, 1, 2], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0013","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aba1b76340649377d3a704504dceeec35b7eb318185c3b833659f5daaae8dc0a","family_id":"qf-sequence-prefix","seed":202609305213,"semantic_template":"qf-sequence-prefix/{\"L\":5,\"d\":2,\"operation\":\"prefix\",\"u\":0}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(0)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.","answer":{"kind":"short_text","value":"[0, 2, 4, 6, 8]","explanation":"Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [0, 2, 4, 6, 8], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0014","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93900944278c9302af7869b33fe1b2367df280ca1d669be644ab8d651f59e974","family_id":"qf-sequence-prefix","seed":202609305214,"semantic_template":"qf-sequence-prefix/{\"L\":6,\"d\":3,\"operation\":\"prefix\",\"u\":1}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(1)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.","answer":{"kind":"short_text","value":"[1, 4, 7, 10, 13, 16]","explanation":"Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [1, 4, 7, 10, 13, 16], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0015","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51e862196097443c9c35722ee85fc5d890bf8173ee8323ce69ebc5c569c01b0f","family_id":"qf-sequence-prefix","seed":202609305215,"semantic_template":"qf-sequence-prefix/{\"L\":7,\"d\":-3,\"operation\":\"prefix\",\"u\":2}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(2)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.","answer":{"kind":"short_text","value":"[2, -1, -4, -7, -10, -13, -16]","explanation":"Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [2, -1, -4, -7, -10, -13, -16], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0016","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"282996731c925a7570205946cdf64aace563767bb11da3a0ffde8ae6388a5173","family_id":"qf-sequence-prefix","seed":202609305216,"semantic_template":"qf-sequence-prefix/{\"L\":8,\"d\":-2,\"operation\":\"prefix\",\"u\":3}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(3)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.","answer":{"kind":"short_text","value":"[3, 1, -1, -3, -5, -7, -9, -11]","explanation":"Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [3, 1, -1, -3, -5, -7, -9, -11], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0017","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"242203aedf2a4d99fa70055c5e3c0e6048972e629aff6bd201f789acda4b9130","family_id":"qf-sequence-prefix","seed":202609305217,"semantic_template":"qf-sequence-prefix/{\"L\":1,\"d\":-1,\"operation\":\"prefix\",\"u\":4}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(4)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.","answer":{"kind":"short_text","value":"[4]","explanation":"Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [4], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0018","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a96c26da6d13141bdb0fa429a6b46362bde1aef0c6f96d478451fa228d271cc","family_id":"qf-sequence-prefix","seed":202609305218,"semantic_template":"qf-sequence-prefix/{\"L\":2,\"d\":0,\"operation\":\"prefix\",\"u\":5}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(5)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.","answer":{"kind":"short_text","value":"[5, 5]","explanation":"Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [5, 5], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0019","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d6b341d9d0d2e5beca18ed72da2c38be7601baf992f36c2afb366bca102cac","family_id":"qf-sequence-prefix","seed":202609305219,"semantic_template":"qf-sequence-prefix/{\"L\":3,\"d\":1,\"operation\":\"prefix\",\"u\":6}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(6)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.","answer":{"kind":"short_text","value":"[6, 7, 8]","explanation":"Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [6, 7, 8], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0020","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"974c9b329dfacf2073b5b00cab59e3cb5d27b00019df3cdb3a494398aff9a69e","family_id":"qf-sequence-prefix","seed":202609305220,"semantic_template":"qf-sequence-prefix/{\"L\":4,\"d\":2,\"operation\":\"prefix\",\"u\":7}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.","answer":{"kind":"short_text","value":"[7, 9, 11, 13]","explanation":"Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [7, 9, 11, 13], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0021","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a956538691cf98a0f4a1344991dea6d9a3e2ac3d1013732e17c695c0cc8c37ea","family_id":"qf-sequence-prefix","seed":202609305221,"semantic_template":"qf-sequence-prefix/{\"L\":5,\"d\":3,\"operation\":\"prefix\",\"u\":8}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(8)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.","answer":{"kind":"short_text","value":"[8, 11, 14, 17, 20]","explanation":"Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [8, 11, 14, 17, 20], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0022","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7196f49c3641f799035b253e6c165ce5d13930aee207c979c1ecf7bc92663729","family_id":"qf-sequence-prefix","seed":202609305222,"semantic_template":"qf-sequence-prefix/{\"L\":6,\"d\":-3,\"operation\":\"prefix\",\"u\":9}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(9)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.","answer":{"kind":"short_text","value":"[9, 6, 3, 0, -3, -6]","explanation":"Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [9, 6, 3, 0, -3, -6], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0023","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78caa911cde872692bb6c41c637531efc3f7a00ed49839070c8e8b2342e6aad9","family_id":"qf-sequence-prefix","seed":202609305223,"semantic_template":"qf-sequence-prefix/{\"L\":7,\"d\":-2,\"operation\":\"prefix\",\"u\":10}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(10)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.","answer":{"kind":"short_text","value":"[10, 8, 6, 4, 2, 0, -2]","explanation":"Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [10, 8, 6, 4, 2, 0, -2], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0024","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b523c5b1ca0dd0a0dd57ac9abd4312d4835352ce6758e3f4ef7cf3f896a55594","family_id":"qf-sequence-prefix","seed":202609305224,"semantic_template":"qf-sequence-prefix/{\"L\":8,\"d\":-1,\"operation\":\"prefix\",\"u\":11}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(11)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.","answer":{"kind":"short_text","value":"[11, 10, 9, 8, 7, 6, 5, 4]","explanation":"Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [11, 10, 9, 8, 7, 6, 5, 4], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-prefix-0025","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f623768dd8d7dfbab87a33d234912660dab88144badaabb6dc76db7ff4839e9","family_id":"qf-sequence-prefix","seed":202609305225,"semantic_template":"qf-sequence-prefix/{\"L\":1,\"d\":0,\"operation\":\"prefix\",\"u\":12}"},"fields":{"title":"Finite Prefix of an Explicit Sequence","stem":"The real sequence is indexed from 0 with a_n=(12)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.","answer":{"kind":"short_text","value":"[12]","explanation":"Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [12], not a guessed continuation rule."},"skills":["prefix length","index order"],"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."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r05-sequence-finite-index-validity-0001","batch_id":"batch-20260930-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5feabc05fa998ed83499178315656f9e6ee008cd32bc3d404af1314278d5adb2","family_id":"qf-sequence-finite-index-validity","seed":202609305301,"semantic_template":"qf-sequence-finite-index-validity/{\"L\":0,\"n\":-1,\"operation\":\"finite_index\"}"},"fields":{"title":"Index Boundary of a Finite Sequence","stem":"A finite real sequence has length L=0 and zero-based valid indices satisfying 0≤n