Quadratic odd-difference growth with one mismatch — pick the odd term Sequence: 19, 28, 39, 52, 67, 84, 102. Which number is the odd one out?

Introduction / Context:
Sequences that grow by successive odd numbers are common. If the odd increments are expected to increase by 2 each time (e.g., +9, +11, +13, +15, +17, +19, ...), a single term violating the next required jump will be the odd one out.

Given Data / Assumptions:

• Series: 19, 28, 39, 52, 67, 84, 102.
• Differences: 28−19=9, 39−28=11, 52−39=13, 67−52=15, 84−67=17, 102−84=18.

Concept / Approach:
The pattern of differences is intended to be consecutive odd numbers: 9, 11, 13, 15, 17, 19, ... Therefore after 84, the next value should be 84 + 19 = 103. A value of 102 indicates a break in that rule and is thus the misfit.

Step-by-Step Solution:
Compute differences: 9, 11, 13, 15, 17 (all odd, increasing by 2).Expected next difference = 19.Expected next term = 84 + 19 = 103.Given next term = 102 (short by 1) ⇒ mismatch.

Verification / Alternative check:
If we replace 102 with 103, the differences become 9, 11, 13, 15, 17, 19 — a perfect consecutive odd-number pattern. Hence 102 is the sole inconsistency.

Why Other Options Are Wrong:
52, 84, 67, and 39 fit the consecutive odd-difference growth; removing any of them breaks earlier correct steps.

Common Pitfalls:
Assuming constant differences, or missing that the gap sequence is itself structured (odd integers increasing by 2). Always analyze the first few differences to detect intent.

Final Answer:
102