Find the missing term in the sequence: 7, 8, 18, ?, 232, 1165. Use the pattern a_{k+1} = a_k * k + k.

Introduction / Context:
A frequently used constructive rule is “multiply by the step index and add the same index.” We test this to fill the blank.

Given Data / Assumptions:

• Series: 7, 8, 18, ?, 232, 1165.
• Index k starts at 1 for the transition 7 → 8.

Concept / Approach:
Apply a_{k+1} = a_k * k + k for k = 1, 2, 3, … and see if all given terms align.

Step-by-Step Solution:

Verification / Alternative check:
The consistent rule fits all observed transitions uniquely, confirming the missing value.

Why Other Options Are Wrong:
36, 42, 84 do not satisfy the recurrence and break the subsequent terms 232 and 1165.

Common Pitfalls:
Using a fixed multiplier; note the multiplier grows with the step index.

Final Answer:
57.