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Sums with sigma notation

Problem 4.1472 · easy

Evaluate \( \displaystyle \sum_{i=1}^{15} \left(- 5 i - 3\right) \) 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
\[ \left(-5\right) 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} - 3 \cdot 15 = -645 \]
From i = 1 to 15.✓ Proved
\[ \sum_{i=1}^{15} \left(- 5 i - 3\right) = -645 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -645 \)

✓ 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 specific bounds and terms. The algebraic steps are verified, and the final result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly identifies the necessary summation formulas and applies them to the specific bounds and terms. The algebraic steps are verified, and the final result is correct.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the necessary summation formulas and applies them to the given arithmetic series. The calculation is verified by direct evaluation.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies the linearity of summation and the standard formulas for the sum of integers and constants. The arithmetic is correct, and the direct evaluation confirms the result.
gpt-oss:20b: pass 2026-11-07

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 2026-11-08 with SymPy 1.14.0.