Sums with sigma notation
Problem 4.1235 · medium
Evaluate \( \displaystyle \sum_{i=2}^{15} \left(- 2 i^{2} - 4 i + 6\right) \) 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 15 and subtract the sum from 1 to 1.
- \[ \left(-2\right) 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} - 4 \cdot 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} + 6 \cdot 15 = -2870 \]From i = 1 to 15.✓ Proved
- \[ \sum_{i=1}^{1} \left(- 2 i^{2} - 4 i + 6\right) = 0 \]From i = 1 to 1.✓ Proved
- \[ -2870 + 0 = -2870 \]Subtract.✓ Proved
- \[ \sum_{i=2}^{15} \left(- 2 i^{2} - 4 i + 6\right) = -2870 \]The sum, evaluated directly, agrees.✓ Proved
Answer \( -2870 \)
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 computes the sum from 1 to 15 but fails to subtract the term for i=1. Since the term at i=1 is 0, the numerical result is coincidentally correct, but the method described (subtracting the sum from 1 to 1) is not executed in the arithmetic of step 5, which simply adds 0 rather than demonstrating the subtraction logic. More critically, step 5 claims '-2870 + 0 = -2870' is the result of 'Subtract', but the previous line established the sum from 1 to 1 is 0. The logic is flawed because it implies the sum from 2 to 15 is Sum(1 to 15) - Sum(1 to 1), but step 5 just adds the result of Sum(1 to 1) (which is 0) to the total, effectively doing nothing. While the number is right, the explanation of the subtraction step is mathematically vacuous and potentially confusing regarding the bounds.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution computes the sum from 1 to 15 but fails to subtract the term for i=1. Since the term at i=1 is 0, the numerical result is coincidentally correct, but the method described (subtracting the sum from 1 to 1) is not executed in the arithmetic of step 5, which simply adds 0 rather than demonstrating the subtraction logic. More critically, step 5 claims '-2870 + 0 = -2870' is the result of 'Subtract', but the previous line established the sum from 1 to 1 is 0. The logic is flawed because it implies the sum from 2 to 15 is Sum(1 to 15) - Sum(1 to 1), but step 5 just adds the result of Sum(1 to 1) (which is 0) to the total, effectively doing nothing. While the number is right, the explanation of the subtraction step is mathematically vacuous and potentially confusing regarding the bounds.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution calculates the sum from i=1 to 15 but fails to subtract the term for i=1. Since the term at i=1 is 0, the numerical answer happens to be correct, but the method described (subtracting the sum from 1 to 1) is not executed in the arithmetic steps, making the logic incomplete and potentially misleading if the first term were non-zero.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to subtract the term for i=1. The term at i=1 is -2(1)^2 - 4(1) + 6 = 0, so the numerical result is coincidentally correct, but the logic in step 5 ('-2870 + 0') implies adding the i=1 term rather than subtracting it, or ignores the subtraction entirely. A correct application of the method described in step 2 requires calculating Sum(1 to 15) - Sum(1 to 1). Since Sum(1 to 1) is 0, the result is -2870 - 0 = -2870. However, step 5 writes '-2870 + 0', which is algebraically equivalent but logically inconsistent with the 'subtract' instruction if interpreted as 'Total - FirstTerm'. More critically, if the first term were non-zero, this method would fail. The primary error is that the solution does not explicitly perform the subtraction of the i=1 term in the final calculation step, relying on the fact that the term is zero. While the final number is right, the procedure shown is flawed because it doesn't demonstrate the subtraction mechanism required by the problem statement 'using summation formulas' for a range not starting at 1. Actually, looking closer, step 5 says '-2870 + 0 = -2870 -- Subtract'. This is confusing notation. It should be '-2870 - 0'. But the bigger issue is that step 3 calculates the sum from 1 to 15. Step 4 calculates the sum from 1 to 1. Step 5 combines them. The combination is labeled 'Subtract' but uses a plus sign. This is a sign error in the notation, even if the value 0 makes it numerically harmless. It teaches bad habits.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 2026-10-19 with SymPy 1.14.0.