Find the next number in the series: 1, 7, 8, 49, 50, 56, 57, 343, 344, 350,…

Concept: This is a rule-generated (not fixed-difference) number series. A single rule produces every new term from an already-existing term in the series: multiply that term by 7, and separately, take that same product and add 1 to it. Applying the rule to every term produced in one round creates the next round of terms, and the first value of each new round always equals 7 raised to one more power than the previous round's first value.

Application:

1. Round 1 (seed = 1): 1 × 7 = 7, and 7 + 1 = 8 → terms 7, 8.

2. Round 2 (apply the rule to 7 and 8, in order): 7 × 7 = 49 → 49 + 1 = 50; 8 × 7 = 56 → 56 + 1 = 57 → terms 49, 50, 56, 57.

3. Round 3 (apply the rule to 49, 50, 56, 57, in order): 49 × 7 = 343 → 343 + 1 = 344; 50 × 7 = 350 → 350 + 1 = 351; 56 × 7 = 392 → 392 + 1 = 393; 57 × 7 = 399 → 399 + 1 = 400 → terms 343, 344, 350, 351, 392, 393, 399, 400.

4. Round 4 (the term being asked for): apply the rule to the first term of round 3, i.e. 343: 343 × 7 = 2401 — this is the next number in the series.

Cross-check: the first term opening each round — 1, 7, 49, 343 — are successive powers of 7 (70, 71, 72, 73); the next power in that backbone is 74 = 2401, which independently confirms the round-4 result above.