One term in the number series is wrong. Find out the wrong term? 325, 259,…

Concept — In a “find the wrong term” number series, every printed term is meant to come from one single rule. The standard first probe is the sequence of first differences: subtract each term from the term before it. When those differences themselves follow a simple regular pattern, the differences that still obey it reveal the rule, and the mistyped term shows up as a break: a faulty term lying inside the series breaks exactly the two differences that touch it — the one entering that term and the one leaving it — while a faulty first or last term would break only the single difference at that end. Once the rule has been read off from the untouched differences, it also tells you the value the faulty term should have carried.

Application — Take the first differences of 325, 259, 202, 160, 127, 105, 94:

1. 325 − 259 = 66

2. 259 − 202 = 57

3. 202 − 160 = 42

4. 160 − 127 = 33

5. 127 − 105 = 22

6. 105 − 94 = 11

Note where each of these values sits. The differences 66, 33, 22 and 11 occupy the 1st, 4th, 5th and 6th places of this list, and the descending chain of multiples of 11 that begins at 66 and ends at 11 across six places is 66, 55, 44, 33, 22, 11, each value 11 less than the one before it. The four observed values match that chain place for place, so it is the intended rule. The two differences that do not match it, 57 and 42, stand next to each other, and the term they enclose is 202. Applying the rule at that place gives 259 − 55 = 204 and then 204 − 44 = 160, so the term printed as 202 should have been 204.

Cross-check — Rebuild the whole series from 325 using the differences 66, 55, 44, 33, 22, 11:

Only the term at position 3 differs (202 against 204); the other six terms match exactly, which agrees with the statement that just one term of the series is faulty. A second confirmation: 57 + 42 = 99 and 55 + 44 = 99, so the two broken differences still total the same as the two intact ones — the fingerprint of a single displaced term (202 is 2 below 204, so one difference gains 2 and the next loses 2) rather than a change of rule.

The term of the series that does not follow the rule is therefore 202, which should have been 204.