Sums with sigma notation
Problem 4.1474 · medium
Evaluate \( \displaystyle \sum_{i=5}^{12} \left(i^{3} + 2 i^{2} + 3\right) \) 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.Reviewed
- The sum starts at i = 5: compute from 1 to 12 and subtract the sum from 1 to 4.Reviewed
- \[ 3 \cdot 12 + 2 \cdot 12 \cdot 13 \cdot 25 \cdot 1 \cdot \frac{1}{6} + \left(12 \cdot 13 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 7420 \]From i = 1 to 12.✓ Proved
- \[ \sum_{i=1}^{4} \left(i^{3} + 2 i^{2} + 3\right) = 172 \]From i = 1 to 4.✓ Proved
- \[ -172 + 7420 = 7248 \]Subtract.✓ Proved
- \[ \sum_{i=5}^{12} \left(i^{3} + 2 i^{2} + 3\right) = 7248 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 7248 \)
✓ 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 standard formulas for sums of powers. The method of computing the sum from 1 to 12 and subtracting the sum from 1 to 4 is valid and correctly executed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies the linearity of summation and the standard formulas for sums of powers. The method of computing the sum from 1 to 12 and subtracting the sum from 1 to 4 is valid and correctly executed.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-07 — 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 4 is valid and correctly executed.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-07 — 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 4 is valid and correctly executed.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.