Find out the wrong number in the sequence. 52, 51, 48, 43, 34, 27, 16

Concept — a “wrong number” series is built by a single generating rule: either a fixed pattern running through the successive differences, or one closed-form expression in the position of the term. The method is always the same — read the rule off the terms that agree with one another, then find the one term that rule does not produce. Difference runs in these items are usually simple: consecutive odd numbers, consecutive squares, a constant step, or a constant ratio.

Application — the given list. Take the successive differences of 52, 51, 48, 43, 34, 27, 16:

1. 52 − 51 = 1

2. 51 − 48 = 3

3. 48 − 43 = 5

4. 43 − 34 = 9

5. 34 − 27 = 7

6. 27 − 16 = 11

The run reads 1, 3, 5, 9, 7, 11. The first three differences and the last one are consecutive odd numbers taken in order, so the intended run is 1, 3, 5, 7, 9, 11. Only the fourth and fifth differences are out of place, and both of them are measured against the same term — the fifth term of the list. Applying the intended step there gives 43 − 7 = 36, so that position should hold 36 rather than 34, after which 36 − 9 = 27 and 27 − 11 = 16 both follow.

Cross-check — closed form. The same rule can be written in one expression. Counting positions from k = 0, each term is 52 minus a perfect square:

Every position agrees with the closed form except k = 4, where the rule produces 36 while the list shows 34 — a shortfall of 2. The terms on either side, 43 at k = 3 and 27 at k = 5, both match the rule exactly, which confirms the rule is intact throughout and that exactly one term departs from it.

Result — the wrong number in the sequence is 34; the term standing in that position should be 36.