Sums with sigma notation
Problem 4.1158 · medium
Evaluate \( \displaystyle \sum_{i=2}^{10} \left(- i\right) \) 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.
- The sum starts at i = 2: compute from 1 to 10 and subtract the sum from 1 to 1.
- \[ \left(-1\right) 10 \cdot 11 \cdot 1 \cdot \frac{1}{2} = -55 \]From i = 1 to 10.✓ Proved
- \[ \sum_{i=1}^{1} - i = -1 \]From i = 1 to 1.✓ Proved
- \[ -55 + 1 = -54 \]Subtract.✓ Proved
- \[ \sum_{i=2}^{10} - i = -54 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( -54 \)
Lines: 4 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | 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: fail (error) — The solution incorrectly lists formulas for sums of squares and cubes, which are not used and distract from the actual linear sum. More critically, the logic in step 5 is flawed: to subtract the i=1 term (-1) from the total sum (-55), one must perform -55 - (-1) = -54, but the equation shows -55 + 1, implying the term was added rather than subtracted, or confusing the sign of the term being removed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly lists formulas for sums of squares and cubes, which are not used and distract from the actual linear sum. More critically, the logic in step 5 is flawed: to subtract the i=1 term (-1) from the total sum (-55), one must perform -55 - (-1) = -54, but the equation shows -55 + 1, implying the term was added rather than subtracted, or confusing the sign of the term being removed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims to use summation formulas but then computes the sum directly in step 3 without showing the application of the formula. More critically, step 5 performs the arithmetic incorrectly: to subtract the term for i=1 (which is -1) from the total sum (-55), one must calculate -55 - (-1) = -54. The solution writes '-55 + 1', which is arithmetically correct for the result, but the logic described in step 2 ('subtract the sum from 1 to 1') implies subtracting -1, so the operation should be explicitly -55 - (-1). While the final number is right, the explanation of the subtraction step is confusingly phrased as 'Subtract' while adding 1. However, the bigger issue is that step 1 lists formulas for i^2 and i^3 which are irrelevant and misleading for a linear sum. The prompt asks to evaluate using summation formulas; listing irrelevant higher-power formulas is poor practice/misleading.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the summation formulas and handles the change of bounds by subtracting the first term. The arithmetic is correct.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.