In the following questions, a series is given in which one term is wrong with…

Concept

In a "wrong-term" number series, each term is built from the one before it by a single progressive rule of the form term = (previous term × m) + c, where the multiplier m and the addend c both grow by a fixed step as you move along the series. The one value that does not fit this rule is the wrong term. To build the "new series", you begin at that wrong value and apply the same progressive rule from its very first step.

Application

1. Test the rule ×m + c with m and c each starting at 1 and increasing by 1 at every step, beginning from the first term 1.

2. 1 × 1 + 1 = 2, but the series shows 3 in this position, so the position should be 2 and 3 is the wrong term.

3. Verify the rest with the rule: 2 × 2 + 2 = 6, 6 × 3 + 3 = 21, 21 × 4 + 4 = 88, 88 × 5 + 5 = 445, 445 × 6 + 6 = 2676. Every later term fits, which confirms the single wrong value is 3.

4. Now form the new series starting from the wrong term 3, applying the same rule from its first step (×1 + 1, then ×2 + 2, then ×3 + 3, …).

5. I term = 3; II term = 3 × 1 + 1 = 4; III term = 4 × 2 + 2 = 10; IV term = 10 × 3 + 3 = 33.

Cross-check

Take one more step to be sure the pattern is self-consistent: 33 × 4 + 4 = 136, which keeps the same growth shape as the corrected original series (each step multiplies by the next integer and adds that same integer). So the IVth term of the new series is 33.