Sums with sigma notation
Problem 4.1042 · medium
Evaluate \( \displaystyle \sum_{i=5}^{15} \left(- 2 i^{2} + 4 i + 6\right) \) 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 = 5: compute from 1 to 15 and subtract the sum from 1 to 4.
- \[ \left(-2\right) 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} + 6 \cdot 15 + 4 \cdot 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} = -1910 \]From i = 1 to 15.✓ Proved
- \[ \sum_{i=1}^{4} \left(- 2 i^{2} + 4 i + 6\right) = 4 \]From i = 1 to 4.✓ Proved
- \[ -1910 - 4 = -1914 \]Subtract.✓ Proved
- \[ \sum_{i=5}^{15} \left(- 2 i^{2} + 4 i + 6\right) = -1914 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( -1914 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution claims to use summation formulas but the equation in line 3 is algebraically incorrect for the sum from 1 to 15 (the term for the constant 6 is missing the factor of N, and the quadratic term is missing the factor of -2). While the final numerical result is correct, the intermediate step shown is false and misleading.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims to use summation formulas but the equation in line 3 is algebraically incorrect for the sum from 1 to 15 (the term for the constant 6 is missing the factor of N, and the quadratic term is missing the factor of -2). While the final numerical result is correct, the intermediate step shown is false and misleading.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution lists the formula for the sum of cubes (Σ i³) in step 1, but the problem only involves terms up to i². While not strictly an error in cgpt-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.