Limits of sequences
Problem 7.500 · easy
Write the first four terms of \( \displaystyle a_n = \frac{4 n^{2} - 1}{2 n^{2} + n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}1\\\frac{3}{2}\\\frac{5}{3}\\\frac{7}{4}\end{matrix}\right] \]The first four terms.✓ Proved
- Divide top and bottom by n².Reviewed
- \[ \lim_{n \to \infty}\left(\frac{4 n^{2} - 1}{2 n^{2} + n}\right) = 2 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 1, \frac{3}{2}, \frac{5}{3}, \frac{7}{4};\ 2 \)
✓ 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 computes the first four terms and applies the standard method for finding the limit of a rational sequence by dividing by the highest power of n. The logic is sound and the result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the first four terms and applies the standard method for finding the limit of a rational sequence by dividing by the highest power of n. The logic is sound and the result is correct.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the first four terms and applies the standard method for finding the limit of a rational sequence by dividing by the highest power of n. The result is correct.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.