Home›Calculus 1›Sums with sigma notation›Problem 4.1301

Sums with sigma notation

Problem 4.1301 · easy

Evaluate \( \displaystyle \sum_{i=1}^{10} \left(- i^{2} + 2 i - 1\right) \) 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
\[ \left(-1\right) 10 \cdot 11 \cdot 21 \cdot 1 \cdot \frac{1}{6} - 1 \cdot 10 + 2 \cdot 10 \cdot 11 \cdot 1 \cdot \frac{1}{2} = -285 \]
From i = 1 to 10.✓ Proved
\[ \sum_{i=1}^{10} \left(- i^{2} + 2 i - 1\right) = -285 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -285 \)

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 standard summation formulas to evaluate the finite sum. The algebraic steps are verified, and the final result matches direct computation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies standard summation formulas to evaluate the finite sum. The algebraic steps are verified, and the final result matches direct computation.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the summation formulas and applies them to the specific terms. The arithmetic in the equation line is correct, and the final result matches the direct evaluation.
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 lists the formula for the sum of cubes, which is not used in the problem. More critically, the equation in line 2 is algebraically inco
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.