Sums with sigma notation
Problem 4.1161 · medium
Evaluate \( \displaystyle \sum_{i=3}^{30} \left(3 i^{2} - 3 i - 3\right) \) 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 = 3: compute from 1 to 30 and subtract the sum from 1 to 2.
- \[ \left(-3\right) 30 \cdot 31 \cdot 1 \cdot \frac{1}{2} - 3 \cdot 30 + 3 \cdot 30 \cdot 31 \cdot 61 \cdot 1 \cdot \frac{1}{6} = 26880 \]From i = 1 to 30.✓ Proved
- \[ \sum_{i=1}^{2} \left(3 i^{2} - 3 i - 3\right) = 0 \]From i = 1 to 2.✓ Proved
- \[ 0 + 26880 = 26880 \]Subtract.✓ Proved
- \[ \sum_{i=3}^{30} \left(3 i^{2} - 3 i - 3\right) = 26880 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 26880 \)
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 incorrectly calculates the sum from 1 to 30 by omitting the summation of the constant term (-3), which should contribute -3*30 = -90. Additionally, the subtraction logic in step 5 is flawed: it adds 0 instead of subtracting the sum from 1 to 2 (which is actually -6, not 0), leading to an incorrect final result.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly calculates the sum from 1 to 30 by omitting the summation of the constant term (-3), which should contribute -3*30 = -90. Additionally, the subtraction logic in step 5 is flawed: it adds 0 instead of subtracting the sum from 1 to 2 (which is actually -6, not 0), leading to an incorrect final result.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims to subtract the sum from 1 to 2, but Equation 5 adds 0 to 26880 instead of subtracting it. Although the term for i=1 to 2 happens to be 0, making the numerical result correct, the logical operation described ('Subtract') does not match the equation shown ('0 + 26880'), which is misleading and technically incorrect arithmetic representation of the subtraction step.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the method of splitting the summation range and uses the standard summation formulas. The arithmetic checks out, and the final result is correct.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.