Limits of sequences
Problem 7.374 · easy
Write the first four terms of \( \displaystyle a_n = - n + \sqrt{n^{2} + 5 n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}-1 + \sqrt{6}\\-2 + \sqrt{14}\\-3 + 2 \sqrt{6}\\2\end{matrix}\right] \]The first four terms.✓ Proved
- Multiply by the conjugate.Reviewed
- \[ \lim_{n \to \infty}\left(- n + \sqrt{n^{2} + 5 n}\right) = \frac{5}{2} \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = -1 + \sqrt{6}, -2 + \sqrt{14}, -3 + 2 \sqrt{6}, 2;\ \frac{5}{2} \)
✓ 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 conjugate method to find the limit. The steps are logically sound and the result is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the first four terms and applies the standard conjugate method to find the limit. The steps are logically sound and the result is correct.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly calculates the first four terms and applies the standard conjugate method to find the correct limit.gpt-oss:20b: pass 2026-10-06 — The solution correctly lists the first four terms and correctly computes the limit. The sentence “Multiply by the conjugate” is unnecessary and lacks detail, but it does not introduce any error. It could be omitted or expanded for clarity.
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 2026-10-06 with SymPy 1.14.0.