Number series – insert the missing term: 2, 8, 18, ?, 50, 72

Introduction / Context:
This is a pattern-recognition series. Observing values against simple polynomial forms (squares, multiples) often reveals the rule efficiently.

Observation / Approach:

• Check 2n^2: for n = 1..6 we get 2, 8, 18, 32, 50, 72 — exactly the given positions with one missing term.
• Thus a_n = 2n^2 fits perfectly.

Step-by-Step Solution:
n=1 → 2; n=2 → 8; n=3 → 18; n=4 → 32; n=5 → 50; n=6 → 72.The missing fourth term is 32.

Verification / Alternative check:
Second differences are constant: first differences 6, 10, 14, 18, 22; second differences all 4 — confirming a quadratic with leading coefficient 2.

Why Other Options Are Wrong:
33, 36, 38, 30 break the exact 2n^2 rule and disrupt constant second differences.

Common Pitfalls:
Overfitting with ad-hoc difference jumps instead of recognizing a simple quadratic pattern.

Final Answer:
32