Sums with sigma notation
Problem 4.1466 · medium
Evaluate \( \displaystyle \sum_{i=2}^{30} \left(i^{3} - 3 i^{2} - 2\right) \) 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.
- The sum starts at i = 2: compute from 1 to 30 and subtract the sum from 1 to 1.
- \[ \left(-3\right) 30 \cdot 31 \cdot 61 \cdot 1 \cdot \frac{1}{6} - 2 \cdot 30 + \left(30 \cdot 31 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 187800 \]From i = 1 to 30.✓ Proved
- \[ \sum_{i=1}^{1} \left(i^{3} - 3 i^{2} - 2\right) = -4 \]From i = 1 to 1.✓ Proved
- \[ 4 + 187800 = 187804 \]Subtract.✓ Proved
- \[ \sum_{i=2}^{30} \left(i^{3} - 3 i^{2} - 2\right) = 187804 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 187804 \)
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: fail (misleading) — Sentence 2 incorrectly states that the sum from 1 to 1 should be subtracted, yet the calculation adds 4 (the negative of that sum). The wording misleads a student about the correct operation.qwen3.6:27b-mlx: pass — The solution correctly applies the standard summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The arithmetic and logic are sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies the standard summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The arithmetic and logic are sound.gpt-oss:20b: fail (misleading) 2026-11-08 — Sentence 2 incorrectly states that the sum from 1 to 1 should be subtracted, yet the calculation adds 4 (the negative of that sum). The wording misleads a student about the correct operation.qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies the summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The arithmetic is verified as correct.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: fail (error) 2026-11-07 — Step 5 claims to subtract the i=1 term but performs addition (4 + 187800). Since the i=1 term is -4, subtracting it requires adding 4, so the arithmetic is correct, but the sentence 'Subtract' is misleadingly imprecise regarding the sign. However, the primary error is in Step 3: the formula for sum of cubes is applied as (30*31*(1/2))**2 which equals 22148100, but the term -3*30*31*61/6 is -28845 and -2*30 is -60. 22148100 - 28845 - 60 = 22119195, not 187800. The equation in step 3 is numerically false.gpt-oss:20b: pass 2026-11-07
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 2026-11-08 with SymPy 1.14.0.