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Sums with sigma notation

Problem 4.1233 · medium

Evaluate \( \displaystyle \sum_{i=2}^{15} \left(3 i^{2} - 3 i - 2\right) \) 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.
The sum starts at i = 2: compute from 1 to 15 and subtract the sum from 1 to 1.
\[ \left(-3\right) 15 \cdot 16 \cdot 1 \cdot \frac{1}{2} - 2 \cdot 15 + 3 \cdot 15 \cdot 16 \cdot 31 \cdot 1 \cdot \frac{1}{6} = 3330 \]
From i = 1 to 15.✓ Proved
\[ \sum_{i=1}^{1} \left(3 i^{2} - 3 i - 2\right) = -2 \]
From i = 1 to 1.✓ Proved
\[ 2 + 3330 = 3332 \]
Subtract.✓ Proved
\[ \sum_{i=2}^{15} \left(3 i^{2} - 3 i - 2\right) = 3332 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 3332 \)

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: pass
qwen3.6:27b-mlx: fail (error) — Step 5 claims to subtract the sum from 1 to 1, but performs addition (2 + 3330) instead of subtraction (3330 - (-2)). While the arithmetic result is coincidentally correct because subtracting a negative is addition, the sentence explicitly states 'Subtract' which contradicts the operation shown and the logic of removing the i=1 term.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 5 claims to subtract the sum from 1 to 1, but performs addition (2 + 3330) instead of subtraction (3330 - (-2)). While the arithmetic result is coincidentally correct because subtracting a negative is addition, the sentence explicitly states 'Subtract' which contradicts the operation shown and the logic of removing the i=1 term.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 5 claims to subtract the term for i=1, but performs addition (3330 + 2) instead of subtraction (3330 - (-2)). While the arithmetic result 3332 is correct, the described operation is wrong.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 5 incorrectly adds the i=1 term instead of subtracting it. To evaluate the sum from i=2 to 15, one must compute Sum(1 to 15) - Sum(1 to 1). Since Sum(1 to 1) is -2, the calculation should be 3330 - (-2) = 3332. The text says '2 + 3330', which implies adding the absolute value or mislabeling the operation, but the logic 'Subtract' in the comment contradicts the arithmetic shown if interpreted as subtracting the term itself. More critically, Step 3 calculates the sum from 1 to 15 as 3330. Let's verify: Sum(3i^2 - 3i - 2) from 1 to 15. Sum(3i^2) = 3 * 15*16*31/6 = 3 * 1240 = 3720. Sum(-3i) = -3 * 15*16/2 = -360. Sum(-2) = -2 * 15 = -30. Total = 3720 - 360 - 30 = 3330. Correct. Sum(1 to 1) is 3(1)^2 - 3(1) - 2 = -2. Correct. Sum(2 to 15) = 3330 - (-2) = 3332. The solution writes '2 + 3330 = 3332'. This is numerically correct but logically confusing because it adds 2 instead of subtracting -2. However, the comment says '-- Subtract'. This is a contradiction between the operation described and the arithmetic shown. A student might think you subtract the value 2, or add the value 2. The correct step is 3330 - (-2). Writing '2 + 3330' obscures the subtraction of the lower bound term. It is misleading.
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.