Find the odd one out: 3, 13, 29, 54, 87, 128

Concept: This is a second-order (quadratic) number series: instead of the terms themselves following a fixed rule, the first differences between consecutive terms follow their own arithmetic progression — the second differences (the differences of the differences) stay constant. The term that breaks a series like this is found by computing the first differences and checking where that constant second difference fails.

Application:

1. List the terms: 3, 13, 29, 54, 87, 128.

2. Compute the first differences between consecutive terms: 13 − 3 = 10, 29 − 13 = 16, 54 − 29 = 25, 87 − 54 = 33, 128 − 87 = 41, giving the run 10, 16, 25, 33, 41.

3. Check whether this run of first differences itself has a constant second difference: 16 − 10 = 6, 25 − 16 = 9, 33 − 25 = 8, 41 − 33 = 8, giving 6, 9, 8, 8 — not constant, so one of the early terms is disturbing the pattern.

4. Test replacing 13 with 12: the first differences become 12 − 3 = 9, 29 − 12 = 17, 54 − 29 = 25, 87 − 54 = 33, 128 − 87 = 41, giving the run 9, 17, 25, 33, 41.

5. Recompute the second differences of this corrected run: 17 − 9 = 8, 25 − 17 = 8, 33 − 25 = 8, 41 − 33 = 8 — constant at 8 across the whole run.

6. Since replacing 13 by 12 is the single change that makes the second difference constant throughout, 13 is the term that disrupts the series.

Cross-check: Extending the corrected run forward, the next first difference would be 41 + 8 = 49, giving a next term of 128 + 49 = 177 — the same constant-second-difference rule continues to hold, confirming the correction.

Result: 13 is the odd one out.