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Sums with sigma notation

Problem 4.1034 · medium

Evaluate \( \displaystyle \sum_{i=3}^{50} \left(- 2 i^{2} + 3 i - 4\right) \) 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 50 and subtract the sum from 1 to 2.
Reviewed
\[ \left(-2\right) 50 \cdot 51 \cdot 101 \cdot 1 \cdot \frac{1}{6} - 4 \cdot 50 + 3 \cdot 50 \cdot 51 \cdot 1 \cdot \frac{1}{2} = -82225 \]
From i = 1 to 50.✓ Proved
\[ \sum_{i=1}^{2} \left(- 2 i^{2} + 3 i - 4\right) = -9 \]
From i = 1 to 2.✓ Proved
\[ -82225 + 9 = -82216 \]
Subtract.✓ Proved
\[ \sum_{i=3}^{50} \left(- 2 i^{2} + 3 i - 4\right) = -82216 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -82216 \)

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the linearity of summation and the method of subtracting the initial terms (1 to 2) from the full sum (1 to 50) to evaluate the sum from 3 to 50. The arithmetic and formulas are correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14

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 2026-10-17 with SymPy 1.14.0.