Limits of sequences
Problem 7.291 · easy
Write the first four terms of \( \displaystyle a_n = - n + \sqrt{n^{2} + 2 n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}-1 + \sqrt{3}\\-2 + 2 \sqrt{2}\\-3 + \sqrt{15}\\-4 + 2 \sqrt{6}\end{matrix}\right] \]The first four terms.✓ Proved
- Multiply by the conjugate.
- \[ \lim_{n \to \infty}\left(- n + \sqrt{n^{2} + 2 n}\right) = 1 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = -1 + \sqrt{3}, -2 + 2 \sqrt{2}, -3 + \sqrt{15}, -4 + 2 \sqrt{6};\ 1 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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: fail (style) — The sentence “Multiply by the conjugate” is vague and does not specify what is being multiplied or why; a clearer explanation of the algebraic manipulation and a brief justification that the limit implies convergence would improve the solution.qwen3.6:27b-mlx: fail (error) — The solution skips the algebraic steps showing how multiplying by the conjugate leads to the limit. It states the method but does not demonstrate the application, leaving the student without the crucial simplification step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution skips the algebraic steps showing how multiplying by the conjugate leads to the limit. It states the method but does not demonstrate the application, leaving the student without the crucial simplification step.gpt-oss:20b: fail (style) 2026-10-05 — The sentence “Multiply by the conjugate” is vague and does not specify what is being multiplied or why; a clearer explanation of the algebraic manipulation and a brief justification that the limit implies convergence would improve the solution.qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the first four terms and the limit. The sentence 'Multiply by the conjugate' accurately describes the standard method used to evaluate the limit, and the final result is correct.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.