{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-sequence"},"kind":"guides","total":2,"pages":1,"page":1,"entries":[{"id":"derivation-mko-sequence-01","title":{"zh-Hant":"數列：由初始條件與遞迴確定通項，以及索引與有限前綴的陷阱","en":"Sequences: fixing the general term from an initial condition and a recurrence, and the pitfalls of indexing and finite prefixes"},"summary":{"zh-Hant":"設實數數列 a:ℕ→ℝ（ℕ 含 0）滿足 a₀=5、a_{n+1}=a_n+3。(1) 證明恰有一個這樣的數列，且 a_n=5+3n；(2) 計算 a₀ 到 a₄ 與 a₁₀，說明 a₀ 是起始項、a₁₀ 是從 1 數起的第 11 個值；(3) 寫出長度 L=4 的有限數列及其索引集合；(4) 說明前綴 5、8、11 不能唯一決定數列。","en":"Let the real sequence a:ℕ→ℝ (with 0∈ℕ) satisfy a₀=5 and a_{n+1}=a_n+3. (1) Prove that exactly one such sequence exists and that a_n=5+3n; (2) compute a₀ to a₄ and a₁₀, explaining that a₀ is the initial term and a₁₀ is the 11th value when counting from 1; (3) write down the finite sequence of length L=4 and its index set; (4) show that the prefix 5, 8, 11 does not determine the sequence."},"mko_ids":["mko-sequence"]},{"id":"derivation-mko-sequence-limit-01","title":{"zh-Hant":"數列極限：倒數模型的精確尾界與 ε-N 證明","en":"Sequence limits: the exact tail cutoff and an ε-N proof for the reciprocal model"},"summary":{"zh-Hant":"對 u(n)=2−3/(n+1)（n∈ℕ0，即 MKO 模型 L=2、c=−3、d=1）：(1) 證明候選尾界 N 對 ε 合法當且僅當 3/(N+1)<ε；(2) 求 ε=1/2 的最小合法 N，並說明 N=5 為何不合法；(3) 以 ε-N 定義證明 u(n)→2；(4) 說明 N 必須依 ε 選取。","en":"For u(n)=2−3/(n+1) with n∈ℕ0 (the MKO model with L=2, c=−3, d=1): (1) prove that a candidate cutoff N is valid for ε exactly when 3/(N+1)<ε; (2) find the least valid N for ε=1/2 and explain why N=5 fails; (3) prove u(n)→2 from the ε-N definition; (4) show that N must be chosen after ε."},"mko_ids":["mko-sequence-limit"]}],"title":"Concept derivation guides"}