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Sums with sigma notation

Problem 4.1163 · medium

Evaluate \( \displaystyle \sum_{i=5}^{12} \left(i^{3} - 3 i^{2} + 2\right) \) 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
The sum starts at i = 5: compute from 1 to 12 and subtract the sum from 1 to 4.
Reviewed
\[ \left(-3\right) 12 \cdot 13 \cdot 25 \cdot 1 \cdot \frac{1}{6} + 2 \cdot 12 + \left(12 \cdot 13 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 4158 \]
From i = 1 to 12.✓ Proved
\[ \sum_{i=1}^{4} \left(i^{3} - 3 i^{2} + 2\right) = 18 \]
From i = 1 to 4.✓ Proved
\[ -18 + 4158 = 4140 \]
Subtract.✓ Proved
\[ \sum_{i=5}^{12} \left(i^{3} - 3 i^{2} + 2\right) = 4140 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 4140 \)

Lines: 4 proved, 2 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	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
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ 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 linearity of summation and the method of subtracting the partial sum from 1 to 4 from the sum from 1 to 12. The formulas used are standard and applied correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the linearity of summation and the method of subtracting the partial sum from 1 to 4 from the sum from 1 to 12. The formulas used are standard and applied correctly.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the method of splitting the summation range and uses the standard summation formulas. The arithmetic and logic are sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer returned a non-object
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.