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Sums with sigma notation

Problem 4.1304 · easy

Evaluate \( \displaystyle \sum_{i=1}^{50} \left(3 i^{2} + 5 i + 3\right) \) 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
\[ 3 \cdot 50 + 5 \cdot 50 \cdot 51 \cdot 1 \cdot \frac{1}{2} + 3 \cdot 50 \cdot 51 \cdot 101 \cdot 1 \cdot \frac{1}{6} = 135300 \]
From i = 1 to 50.✓ Proved
\[ \sum_{i=1}^{50} \left(3 i^{2} + 5 i + 3\right) = 135300 \]
The sum, evaluated directly, agrees.✓ Proved

Answer \( 135300 \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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: pass — The solution correctly applies standard summation formulas to evaluate the series, and the arithmetic is verified by direct computation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies standard summation formulas to evaluate the series, and the arithmetic is verified by direct computation.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the standard summation formulas for constants, linear, and quadratic terms. The arithmetic in the second line is verified as correct, and the final result matches the direct evaluation.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution incorrectly applies the summation formulas. Specifically, it uses the formula for the sum of squares (Σi²) for the constant term (Σ3), and uses the formula for the sum of cubes (Σi³) for the quadratic term (Σ3i²). The correct application should be 3*Σi² + 5*Σi + 3*Σ1.
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.