Sums with sigma notation
Problem 4.1309 · medium
Evaluate \( \displaystyle \sum_{i=3}^{12} \left(3 i^{2} - i + 6\right) \) 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.Reviewed
- The sum starts at i = 3: compute from 1 to 12 and subtract the sum from 1 to 2.Reviewed
- \[ \left(-1\right) 12 \cdot 13 \cdot 1 \cdot \frac{1}{2} + 6 \cdot 12 + 3 \cdot 12 \cdot 13 \cdot 25 \cdot 1 \cdot \frac{1}{6} = 1944 \]From i = 1 to 12.✓ Proved
- \[ \sum_{i=1}^{2} \left(3 i^{2} - i + 6\right) = 24 \]From i = 1 to 2.✓ Proved
- \[ -24 + 1944 = 1920 \]Subtract.✓ Proved
- \[ \sum_{i=3}^{12} \left(3 i^{2} - i + 6\right) = 1920 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 1920 \)
✓ Nihil obstat Lines: 4 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies the linearity of summation and the method of subtracting the initial terms (1 to 2) from the total sum (1 to 12) to evaluate the sum from 3 to 12. The arithmetic and formulas are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies the linearity of summation and the method of subtracting the initial terms (1 to 2) from the total sum (1 to 12) to evaluate the sum from 3 to 12. The arithmetic and formulas are correct.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the linearity of summation and the method of subtracting the initial terms (1 to 2) from the total sum (1 to 12) to handle the lower bound of 3. The arithmetic and final result are correct.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the linearity of summation and the standard formulas for sums of powers. The strategy of computing the sum from 1 to 12 and subtracting the sum from 1 to 2 is valid and correctly executed.gpt-oss:20b: pass 2026-10-20
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.