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Sums with sigma notation

Problem 4.1231 · easy

Evaluate \( \displaystyle \sum_{i=1}^{25} \left(i^{3} + i^{2} - 5 i - 2\right) \) 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
\[ \left(-5\right) 25 \cdot 26 \cdot 1 \cdot \frac{1}{2} - 2 \cdot 25 + 1 \cdot 25 \cdot 26 \cdot 51 \cdot 1 \cdot \frac{1}{6} + \left(25 \cdot 26 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 109475 \]
From i = 1 to 25.✓ Proved
\[ \sum_{i=1}^{25} \left(i^{3} + i^{2} - 5 i - 2\right) = 109475 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 109475 \)

✓ 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 standard summation formulas and applies them to the given bounds. The algebraic evaluation is verified by the CAS, and the final result matches the direct computation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard summation formulas and applies them to the given bounds. The algebraic evaluation is verified by the CAS, and the final result matches the direct computation.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the summation formulas and applies them to the specific bounds. The arithmetic is verified by the algebra system, and the final result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard summation formulas and applies them to the given bounds. The arithmetic verification confirms the stated answer.
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.