Limits of sequences
Problem 7.289 · easy
Write the first four terms of \( \displaystyle a_n = \frac{3 n^{2} + 1}{4 n^{2} + n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}\frac{4}{5}\\\frac{13}{18}\\\frac{28}{39}\\\frac{49}{68}\end{matrix}\right] \]The first four terms.✓ Proved
- Divide top and bottom by n².Reviewed
- \[ \lim_{n \to \infty}\left(\frac{3 n^{2} + 1}{4 n^{2} + n}\right) = \frac{3}{4} \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = \frac{4}{5}, \frac{13}{18}, \frac{28}{39}, \frac{49}{68};\ \frac{3}{4} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the terms evaluated at n = 10⁶, 10⁶ + 1 and 10⁷ |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the first four terms and determines the limit using a standard algebraic method (dividing by the highest power of n). The steps are logically sound and the results are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the first four terms and determines the limit using a standard algebraic method (dividing by the highest power of n). The steps are logically sound and the results are correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the first four terms and applies the standard method (dividing by the highest power of n) to find the limit. The logic is sound and complete.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/sequence_limits, checked 2026-10-05 with SymPy 1.14.0.