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

Concept

In a wrong-term series, every term is generated from the one before it by a fixed rule whose multiplier and added constant grow by 1 at each step. Here the governing rule is: next term = (previous term × n) + n, where n increases as 2, 3, 4, 5, 6, 7. The wrong term is the single value that does not equal what this rule predicts from its neighbours.

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. Confirm the position another way — work back from the next term, 27: we need a number x with x × 3 + 3 = 27, so x = (27 − 3) ÷ 3 = 8.

3. Both directions agree the second term must be 8, so the printed 6 does not fit the rule.

4. With 8 in place the rest of the chain holds: 8 × 3 + 3 = 27, 100 × 5 + 5 = 505, 505 × 6 + 6 = 3036, and 3036 × 7 + 7 = 21259.

Cross-check

The replacement value 8 is fixed independently by two methods — forward from 3 (3 × 2 + 2) and backward from 27 ((27 − 3) ÷ 3) — and both give 8. Among the listed numbers, 6 is the only one whose correction restores the multiply-by-n-then-add-n pattern on both of its sides.

So the wrong term is 6; it should be 8.