Odd Man Out – Product-plus-index pattern check: 7, 8, 18, 57, 228, 1165, 6996

Introduction / Context:
Many constructed series follow a rule of the form “multiply by k, then add k” where k increases step by step. The task is to verify each transition and find the single term that breaks the otherwise consistent pattern.

Given Data / Assumptions:

• Sequence: 7, 8, 18, 57, 228, 1165, 6996
• Hypothesis: a(n+1) = a(n) * n + n for n starting at 1 (or equivalently “×n then +n”).

Concept / Approach:
Test the evolving multiplier/addend by index: 7*1+1=8; 8*2+2=18; 18*3+3=57; next should be 57*4+4 = 232, not 228. After that, 232*5+5=1165; 1165*6+6=6996. Only one value breaks the chain.

Step-by-Step Solution:
7 → 8: 7*1 + 1 = 8 ✔8 → 18: 8*2 + 2 = 18 ✔18 → 57: 18*3 + 3 = 57 ✔57 → (should be) 57*4 + 4 = 232But given term is 228 ✖Continuing: 232*5 + 5 = 1165 ✔; 1165*6 + 6 = 6996 ✔

Verification / Alternative check:
Replacing 228 with 232 restores a perfect application of the rule across all steps.

Why Other Options Are Wrong:
8, 18, 57, 1165 are each exactly of the form “previous*index + index” as demonstrated.

Common Pitfalls:
Searching for a single constant multiplier or additive when the rule evolves with the index.

Final Answer:
228