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Sums with sigma notation

Problem 4.1046 · easy

Evaluate \( \displaystyle \sum_{i=1}^{12} \left(- 3 i^{2} + 4 i + 5\right) \) using summation formulas.

Split the sum and use Σᵢ₌₁ᴺ 1 = N, Σ i = N(N + 1)/2, Σ i² = N(N + 1)(2N + 1)/6, Σ i³ = [N(N + 1)/2]².
Reviewed
\[ \left(-3\right) 12 \cdot 13 \cdot 25 \cdot 1 \cdot \frac{1}{6} + 5 \cdot 12 + 4 \cdot 12 \cdot 13 \cdot 1 \cdot \frac{1}{2} = -1578 \]
From i = 1 to 12.✓ Proved
\[ \sum_{i=1}^{12} \left(- 3 i^{2} + 4 i + 5\right) = -1578 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -1578 \)

Lines: 2 proved, 1 reviewed. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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 added one by one in Python integers

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies standard summation formulas and the arithmetic is verified by the provided equations. The inclusion of the cubic sum formula in the setup is unnecessary but harmless.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies standard summation formulas and the arithmetic is verified by the provided equations. The inclusion of the cubic sum formula in the setup is unnecessary but harmless.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies standard summation formulas and the arithmetic is verified by the algebra system. The inclusion of the unused cubic formula is harmless.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the summation formulas by omitting the coefficient 4 from the linear term (using 5 instead of 4 for the constant te
gpt-oss:20b: pass 2026-10-14

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/summation_formulas, checked 2026-10-17 with SymPy 1.14.0.