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Sums with sigma notation

Problem 4.1302 · easy

Evaluate \( \displaystyle \sum_{i=1}^{30} \left(- 2 i^{2} + 2 i + 4\right) \) 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
\[ \left(-2\right) 30 \cdot 31 \cdot 61 \cdot 1 \cdot \frac{1}{6} + 4 \cdot 30 + 2 \cdot 30 \cdot 31 \cdot 1 \cdot \frac{1}{2} = -17860 \]
From i = 1 to 30.✓ Proved
\[ \sum_{i=1}^{30} \left(- 2 i^{2} + 2 i + 4\right) = -17860 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -17860 \)

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 the standard summation formulas for i, i^2, and constants. The arithmetic in the equation line is verified as correct, and the final result matches the direct evaluation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies the standard summation formulas for i, i^2, and constants. The arithmetic in the equation line is verified as correct, and the final result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution incorrectly applies the summation formula for the constant term, using 4N instead of 4(N+1) (or 4*31). The constant term sum is \sum_{i=1}^{30} 4 = 4 \times 30 = 120, but the formula \sum_{i=1}^N c = cN is correct, wait. Let's re-read carefully. The sentence says \sum 1 = N. The equation uses 4*30. This is correct for \sum_{i=1}^{30} 4. However, the coefficient for the i^2 term is -2. The formula is N(N+1)(2N+1)/6. For N=30, this is 30*31*61/6 = 9455. -2 * 9455 = -18910. The i term is 2 * 30*31/2 = 30*31 = 930. The constant term is 4 * 30 = 120. Sum = -18910 + 930 + 120 = -17860. The calculation is actually correct. Why did I think it was wrong? Ah, I misread the standard formula application in my head. Let's look at the sentence again. It lists formulas. It does not explicitly state the linearity of summation, but implies it by 'Split the sum'. The equation line is marked 'proved' by CAS, so the arithmetic is correct. The sentence lists \sum i^3 which is not used, but that's not an error, just irrelevant info. Is there a subtle error? The problem asks to evaluate using summation formulas. The solution splits and applies them. The result matches. It seems correct. Wait, let me double check the 'proved' line. -2*30*31*61/6 + 4*30 + 2*30*31/2. This corresponds to -2 \sum i^2 + 4 \sum 1 + 2 \sum i. This is correct. The verdict should be pass? No, look at the prompt instructions. 'Find what is wrong'. Is there anything wrong? The sentence lists \sum i^3. This is unnecessary but not wrong. The setup is correct. The conclusion follows. It seems sound. Let me re-read the 'severity' definitions. 'pass' if sound. It is sound. Why would it fail? Maybe the sentence implies \sum_{i=1}^N 1 = N is a formula to be used, but usually \sum_{i=1}^N c = cN. It's equivalent. I will mark it pass.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the summation formula for the constant term, using 4N instead of 4(N+1) (or failing to account for the sum of 1 fro
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.