4, 13, 41, 126, 382, ?

Concept: In many number series, consecutive terms are not linked by a single constant ratio or a single constant difference. Instead, each term can be produced from the one before it by a compound rule — multiplying by a fixed constant and then adding a term that itself grows by a fixed step. To extend such a series correctly, first confirm that this compound rule reproduces every given term, and only then apply the same rule once more to find the missing term.

Applying this rule to the given series 4, 13, 41, 126, 382, ?:

1. 4 × 3 = 12, and 12 + 1 = 13

2. 13 × 3 = 39, and 39 + 2 = 41

3. 41 × 3 = 123, and 123 + 3 = 126

4. 126 × 3 = 378, and 378 + 4 = 382

5. 382 × 3 = 1146, and 1146 + 5 = 1151

Cross-check: across all five transitions above, the multiplier stays fixed at 3 while the number added rises by exactly 1 each time (1, 2, 3, 4, 5) — confirming a consistent compound rule rather than a coincidence limited to the first few terms.

Hence, the next term in the series is 1151.