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Sums with sigma notation

Problem 4.1165 · easy

Evaluate \( \displaystyle \sum_{i=1}^{20} \left(4 - 2 i\right) \) 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.
\[ \left(-2\right) 20 \cdot 21 \cdot 1 \cdot \frac{1}{2} + 4 \cdot 20 = -340 \]
From i = 1 to 20.✓ Proved
\[ \sum_{i=1}^{20} \left(4 - 2 i\right) = -340 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -340 \)

Lines: 2 proved, 1 not checked. 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	Not checked	—	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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The first sentence claims to split the sum and lists several summation formulas, but only Σi and Σ1 are actually needed. It also does not explicitly
qwen3.6:27b-mlx: pass — The solution correctly identifies the necessary summation formulas and applies them to evaluate the sum. The arithmetic is correct, and the final result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the necessary summation formulas and applies them to evaluate the sum. The arithmetic is correct, and the final result matches the stated answer.
gpt-oss:20b: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The first sentence claims to split the sum and lists several summation formulas, but only Σi and Σ1 are actually needed. It also does not explicitly
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the summation formulas for a constant and a linear term. The arithmetic is correct, and the result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the linearity of summation and the standard formula for the sum of the first N integers. The arithmetic is correct, and the result matches the direct evaluation.
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 2026-10-18 with SymPy 1.14.0.