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Sums with sigma notation

Problem 4.1232 · easy

Evaluate \( \displaystyle \sum_{i=1}^{25} \left(- 2 i^{2} - 4 i - 2\right) \) 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
\[ \left(-2\right) 25 \cdot 26 \cdot 51 \cdot 1 \cdot \frac{1}{6} - 4 \cdot 25 \cdot 26 \cdot 1 \cdot \frac{1}{2} - 2 \cdot 25 = -12400 \]
From i = 1 to 25.✓ Proved
\[ \sum_{i=1}^{25} \left(- 2 i^{2} - 4 i - 2\right) = -12400 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -12400 \)

✓ 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. 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 necessary summation formulas and applies them to the given bounds. The intermediate calculation and final verification are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the necessary summation formulas and applies them to the given bounds. The intermediate calculation and final verification are correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies standard summation formulas to evaluate the finite sum, and the arithmetic is verified by direct computation.
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 given bounds. The arithmetic in the unchecked equations is correct, and the direct evaluation confirms the result.
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 2026-10-19 with SymPy 1.14.0.