Limits of sequences
Problem 7.290 · easy
Write the first four terms of \( \displaystyle a_n = - n + \sqrt{n^{2} + 4 n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}-1 + \sqrt{5}\\-2 + 2 \sqrt{3}\\-3 + \sqrt{21}\\-4 + 4 \sqrt{2}\end{matrix}\right] \]The first four terms.✓ Proved
- Multiply by the conjugate.Reviewed
- \[ \lim_{n \to \infty}\left(- n + \sqrt{n^{2} + 4 n}\right) = 2 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = -1 + \sqrt{5}, -2 + 2 \sqrt{3}, -3 + \sqrt{21}, -4 + 4 \sqrt{2};\ 2 \)
✓ 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 identifies the limit. The sentence 'Multiply by the conjugate' correctly describes the standard algebraic technique used to evaluate the limit of the indeterminate form infinity minus infinity, which yields the correct result of 2.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the first four terms and identifies the limit. The sentence 'Multiply by the conjugate' correctly describes the standard algebraic technique used to evaluate the limit of the indeterminate form infinity minus infinity, which yields the correct result of 2.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-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, which is confirmed by the subsequent equation.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.