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Sums with sigma notation

Problem 4.1237 · medium

Evaluate \( \displaystyle \sum_{i=2}^{15} \left(4 - i^{2}\right) \) 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
The sum starts at i = 2: compute from 1 to 15 and subtract the sum from 1 to 1.
Reviewed
\[ \left(-1\right) 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} + 4 \cdot 15 = -1180 \]
From i = 1 to 15.✓ Proved
\[ \sum_{i=1}^{1} \left(4 - i^{2}\right) = 3 \]
From i = 1 to 1.✓ Proved
\[ -1180 - 3 = -1183 \]
Subtract.✓ Proved
\[ \sum_{i=2}^{15} \left(4 - i^{2}\right) = -1183 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -1183 \)

✓ 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 computing the sum from 1 to N and subtracting the term for i=1. The algebraic steps and final result are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the term for i=1. The algebraic steps and final result are correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the terms for indices less than the lower bound. The arithmetic and logic are sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the term for i=1. The arithmetic and logic are sound.
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.