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Sums with sigma notation

Problem 4.1307 · medium

Evaluate \( \displaystyle \sum_{i=3}^{15} \left(- i^{2} - 3 i + 2\right) \) 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
The sum starts at i = 3: compute from 1 to 15 and subtract the sum from 1 to 2.
Reviewed
\[ \left(-1\right) 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} - 3 \cdot 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} + 2 \cdot 15 = -1570 \]
From i = 1 to 15.✓ Proved
\[ \sum_{i=1}^{2} \left(- i^{2} - 3 i + 2\right) = -10 \]
From i = 1 to 2.✓ Proved
\[ -1570 + 10 = -1560 \]
Subtract.✓ Proved
\[ \sum_{i=3}^{15} \left(- i^{2} - 3 i + 2\right) = -1560 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -1560 \)

✓ Nihil obstat Lines: 4 proved, 2 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	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 method of splitting the summation range and uses the standard summation formulas. The arithmetic and logic are sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — 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 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the sum from 1 to k-1. The arithmetic and logic are sound.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the sum from 1 to k-1. The algebraic steps and final result are correct.
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.