Given below is the series in which one number is wrong. Consider this wrong…

Concept

In a single-rule number series each term is produced from the previous term by one fixed operation whose multiplier and added constant both grow by a fixed step. Here the governing rule is: next = previous × k + k, where k = 1, 2, 3, 4, 5, 6 … in turn. The 'wrong number' is the lone term that does not obey this rule; replacing it with the value the rule demands restores a perfectly regular series.

Application — find the wrong number in series 1

Apply next = previous × k + k starting from 1, with k increasing by 1 at every step:

1. k = 1: 1 × 1 + 1 = 2 (the series shows 3, so this term is the odd one out)

2. k = 2: 2 × 2 + 2 = 6 ✓

3. k = 3: 6 × 3 + 3 = 21 ✓

4. k = 4: 21 × 4 + 4 = 88 ✓

5. k = 5: 88 × 5 + 5 = 445 ✓

6. k = 6: 445 × 6 + 6 = 2676 ✓

Every term from 6 onward fits the rule exactly, so the single wrong number is 3 (it should be 2). The question tells us to take this wrong number as A in the second series, so A = 3.

Application — build series 2 and find D

Starting from A = 3, apply the same rule next = previous × k + k with k = 1, 2, 3:

1. A = 3

2. B = A × 1 + 1 = 3 × 1 + 1 = 4

3. C = B × 2 + 2 = 4 × 2 + 2 = 10

4. D = C × 3 + 3 = 10 × 3 + 3 = 33

Therefore D = 33.

Cross-check

Continue the rule one step past D: 33 × 4 + 4 = 136 and 136 × 5 + 5 = 685 — the same chain that generated series 1, confirming the rule is consistent. Stopping one step early gives 10 (the value of C), and multiplying without the added constant gives 10 × 3 = 30; both are tempting near-misses, which is why the disciplined step-by-step application matters.