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Sums with sigma notation

Problem 4.1308 · medium

Evaluate \( \displaystyle \sum_{i=5}^{30} \left(3 i^{2} + 3 i + 4\right) \) 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 = 5: compute from 1 to 30 and subtract the sum from 1 to 4.
\[ 4 \cdot 30 + 3 \cdot 30 \cdot 31 \cdot 1 \cdot \frac{1}{2} + 3 \cdot 30 \cdot 31 \cdot 61 \cdot 1 \cdot \frac{1}{6} = 29880 \]
From i = 1 to 30.✓ Proved
\[ \sum_{i=1}^{4} \left(3 i^{2} + 3 i + 4\right) = 136 \]
From i = 1 to 4.✓ Proved
\[ -136 + 29880 = 29744 \]
Subtract.✓ Proved
\[ \sum_{i=5}^{30} \left(3 i^{2} + 3 i + 4\right) = 29744 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 29744 \)

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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-21 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the method of splitting the summation range and uses the standard summation formulas appropriately. The arithmetic and logic are sound.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the method of computing the sum from 1 to N and subtracting the sum from 1 to k-1. The formulas cited are standard and correctly applied in the subsequent steps.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.