Sums with sigma notation
Problem 4.1036 · medium
Evaluate \( \displaystyle \sum_{i=2}^{15} \left(i^{3} + 3 i^{2} + 4 i + 1\right) \) 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 = 2: compute from 1 to 15 and subtract the sum from 1 to 1.Reviewed
- \[ 1 \cdot 15 + 4 \cdot 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} + 3 \cdot 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} + \left(15 \cdot 16 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 18615 \]From i = 1 to 15.✓ Proved
- \[ \sum_{i=1}^{1} \left(i^{3} + 3 i^{2} + 4 i + 1\right) = 9 \]From i = 1 to 1.✓ Proved
- \[ -9 + 18615 = 18606 \]Subtract.✓ Proved
- \[ \sum_{i=2}^{15} \left(i^{3} + 3 i^{2} + 4 i + 1\right) = 18606 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 18606 \)
✓ 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 summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The logic is sound and the arithmetic is verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The logic is sound and the arithmetic is verified.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the linearity of summation and the standard formulas for power sums. The method of computing the sum from 1 to 15 and subtracting the term for i=1 is valid and correctly executed.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the summation formulas and handles the lower bound adjustment by subtracting the i=1 term. The arithmetic is correct.gpt-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.