Sums with sigma notation
Problem 4.1032 · medium
Evaluate \( \displaystyle \sum_{i=2}^{25} \left(3 i^{2} + i + 5\right) \) 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 25 and subtract the sum from 1 to 1.Reviewed
- \[ 5 \cdot 25 + 1 \cdot 25 \cdot 26 \cdot 1 \cdot \frac{1}{2} + 3 \cdot 25 \cdot 26 \cdot 51 \cdot 1 \cdot \frac{1}{6} = 17025 \]From i = 1 to 25.✓ Proved
- \[ \sum_{i=1}^{1} \left(3 i^{2} + i + 5\right) = 9 \]From i = 1 to 1.✓ Proved
- \[ -9 + 17025 = 17016 \]Subtract.✓ Proved
- \[ \sum_{i=2}^{25} \left(3 i^{2} + i + 5\right) = 17016 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( 17016 \)
✓ 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 25 and subtracting the term for i=1 is valid and correctly executed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — 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 25 and subtracting the term for i=1 is valid and correctly executed.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 sums of powers. The method of computing the sum from 1 to 25 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 method of computing the sum from 1 to N and subtracting the term for i=1. The formulas used are standard and applied correctly.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.