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Sums with sigma notation

Problem 4.1305 · medium

Evaluate \( \displaystyle \sum_{i=2}^{20} \left(- 3 i^{2} + 5 i + 5\right) \) 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.
The sum starts at i = 2: compute from 1 to 20 and subtract the sum from 1 to 1.
\[ \left(-3\right) 20 \cdot 21 \cdot 41 \cdot 1 \cdot \frac{1}{6} + 5 \cdot 20 + 5 \cdot 20 \cdot 21 \cdot 1 \cdot \frac{1}{2} = -7460 \]
From i = 1 to 20.✓ Proved
\[ \sum_{i=1}^{1} \left(- 3 i^{2} + 5 i + 5\right) = 7 \]
From i = 1 to 1.✓ Proved
\[ -7460 - 7 = -7467 \]
Subtract.✓ Proved
\[ \sum_{i=2}^{20} \left(- 3 i^{2} + 5 i + 5\right) = -7467 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( -7467 \)

Lines: 4 proved, 2 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	Not checked	—	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: fail (error) — The calculation in line 3 is incorrect; it omits the quadratic term's coefficient application and the constant term's summation, resulting in -7460 instead of the correct sum of -2535 for i=1 to 20. Consequently, the final answer is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The calculation in line 3 is incorrect; it omits the quadratic term's coefficient application and the constant term's summation, resulting in -7460 instead of the correct sum of -2535 for i=1 to 20. Consequently, the final answer is wrong.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — Line 3 incorrectly applies the summation formulas for the linear and constant terms. The term 5*i should be 5 * [N(N+1)/2] and the term 5 should be 5 * N, but the equation uses 5*20 for the linear term and 5*20*21*(1/2) for the constant term, effectively swapping the formulas for sum(i) and sum(1).
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — Line 3 contains a significant algebraic error in the application of the summation formulas. The term for \(\sum i\) is calculated as \(5 \times 20\) instead of \(5 \times \frac{20(21)}{2}\), and the term for \(\sum 1\) is calculated as \(5 \times 20 \times 21 \times (1/2)\) instead of \(5 \times 20\). Although the final numerical result in Line 3 (-7460) happens to be correct (likely due to a typo in the displayed expression or a lucky cancellation not shown), the displayed equation is mathematically false and misleading.
gpt-oss:20b: fail (error) 2026-10-20 — The computation for the sum from 1 to 20 is incorrect: the term 5*20 is a mistaken duplicate of 5Σi, and the correct 5Σi should be 5*210=1050. This mis‑application of the summation formulas leads to an incorrect intermediate value.

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.