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Sums with sigma notation

Problem 4.1153 · easy

Evaluate \( \displaystyle \sum_{i=1}^{12} \left(3 i^{2} + 3 i\right) \) 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
\[ 3 \cdot 12 \cdot 13 \cdot 1 \cdot \frac{1}{2} + 3 \cdot 12 \cdot 13 \cdot 25 \cdot 1 \cdot \frac{1}{6} = 2184 \]
From i = 1 to 12.✓ Proved
\[ \sum_{i=1}^{12} \left(3 i^{2} + 3 i\right) = 2184 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 2184 \)

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 identifies the summation formulas and applies them to the given bounds. The arithmetic is verified by the algebra system, and the final result matches the direct evaluation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the summation formulas and applies them to the given bounds. The arithmetic is verified by the algebra system, and the final result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution lists the formula for the sum of cubes, which is not used, but fails to list the formula for the sum of constants (Σ1 = N) required for the linear term 3i. Furthermore, the equation in line 2 incorrectly applies the sum of squares formula to the linear term (using 2N+1=25) and omits the calculation for the linear term entirely, making the derivation mathematically incoherent even though the final number is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the summation formulas and applies them to the split terms. The arithmetic in the unchecked equation is correct, and the final result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.