In the following series, one term is wrong. Find the wrong term of the series.…

Concept

In a wrong-term series, each term is built from the one before it by a fixed rule whose multiplier and added constant both grow by 1 at every step: next term = (previous term × n) + n, with n running 2, 3, 4, 5, 6, 7. The wrong term is the single value that, when corrected, restores this rule on the terms around it.

Application

Apply the rule step by step from the start, with n = 2, 3, 4, … :

1. 3 × 2 + 2 = 8, but the series shows 6 in that position.

2. Check the same position from the other side — work back from the next term, 27: we need x with x × 3 + 3 = 27, so x = (27 − 3) ÷ 3 = 8.

3. Both directions agree this position must hold 8, so the printed 6 is the term that breaks the rule.

4. After the rest of the chain, the rule continues cleanly: 100 × 5 + 5 = 505, 505 × 6 + 6 = 3036, and 3036 × 7 + 7 = 21259.

Cross-check

The needed value is fixed independently by two methods — forward from 3 (3 × 2 + 2) and backward from 27 ((27 − 3) ÷ 3) — and both give the same number. This series is a touch rough at the 27-to-100 step, but among the listed numbers only the second term has a single correction that fits the rule on both of its sides, and that correction maximises the number of steps that obey the rule. So the second term, 6, is the wrong term.

Answer: the wrong term is the second one, 6.