Sums with sigma notation
Problem 4.1039 · medium
Evaluate \( \displaystyle \sum_{i=3}^{25} \left(i^{3} - i^{2} + 3 i + 2\right) \) 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.Reviewed
- The sum starts at i = 3: compute from 1 to 25 and subtract the sum from 1 to 2.Reviewed
- \[ \left(-1\right) 25 \cdot 26 \cdot 51 \cdot 1 \cdot \frac{1}{6} + 2 \cdot 25 + 3 \cdot 25 \cdot 26 \cdot 1 \cdot \frac{1}{2} + \left(25 \cdot 26 \cdot 1 \cdot \frac{1}{2}\right)^{2} = 101125 \]From i = 1 to 25.✓ Proved
- \[ \sum_{i=1}^{2} \left(i^{3} - i^{2} + 3 i + 2\right) = 17 \]From i = 1 to 2.✓ Proved
- \[ -17 + 101125 = 101108 \]Subtract.✓ Proved
- \[ \sum_{i=3}^{25} \left(i^{3} - i^{2} + 3 i + 2\right) = 101108 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 101108 \)
✓ 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 standard technique of computing the sum from 1 to N and subtracting the terms for indices 1 to k-1. The arithmetic and logic are sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the standard technique of computing the sum from 1 to N and subtracting the terms for indices 1 to k-1. The arithmetic and logic are sound.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of subtracting the initial terms (1 to 2) from the total sum (1 to 25) to handle the lower bound of 3. The algebraic steps and final result are verified.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the initial terms. The formulas used are standard and correct, and the arithmetic checks out.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.