Round #1098
21 May 2026, 09:40· math· Meta-Llama-3.3-70B-Instruct· r13-clean-hints
- Complexity
- band 0
- ZP / UO / OBS
- 3 / 0 / 1
- Journey weight
- 223
- Lightshift ΔL
- —
Prompt
Given the recursive sequence a_1 = 2 and a_(n+1) = 3 a_n - 1, compute a_4. Give the exact integer.
Zero-point index (what the engine surfaced)
ZPI-ZPI-0027 | 2026-05-21T09:40:17Z
Given the recursive sequence a_1 = 2 and a_(n+1) = 3 a_n - 1, compute a_4. Give the exact integer.
ZERO POINT — 3 entries (structural invariants)
● hsl(7.18, 100%, 50%) class [feature]
● hsl(0.00, 100%, 50%) public [feature]
● hsl(84.10, 100%, 50%) namespace [feature]
UNOBSERVED — 0 entries (implied truths)
OBSERVED — 1 entries (explicit in request)
◉ hsl(175.38, 100%, 27%) = [feature]
COMPLEXITY │ 4 entries │ spread: 0° │ band: LOW
Baseline response (no ZPI)
## Step 1: Identify the initial term and the recursive formula of the sequence.
The initial term is a_1 = 2, and the recursive formula is a_(n+1) = 3a_n - 1.
## Step 2: Calculate the second term of the sequence using the recursive formula.
a_2 = 3a_1 - 1 = 3(2) - 1 = 6 - 1 = 5.
## Step 3: Calculate the third term of the sequence using the recursive formula.
a_3 = 3a_2 - 1 = 3(5) - 1 = 15 - 1 = 14.
## Step 4: Calculate the fourth term of the sequence using the recursive formula.
a_4 = 3a_3 - 1 = 3(14) - 1 = 42 - 1 = 41.
The final answer is: $\boxed{41}$
Enhanced response (with ZPI)
a_4 = 53.
Grader chain
Each grader reads the same baseline + enhanced pair and casts an independent verdict. Disagreement is the framework's own signal — not noise to be averaged away.
V1samba/Meta-Llama-3.3-70B-Instruct
WorseV4qwen2.5-coder-3b-instruct
WorseV5cerebras/qwen-3-235b-a22b-instruct-2507
WorseV7qwen2.5-coder-7b-instruct
WorseV8openai/gpt-4.1
WorseV9anthropic/claude-opus-4-7
WorseV12openai/gpt-4o
WorseComments
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