Complete the series: 24, 60, 120, 210, ?

Concept:

Some number series are not built from a single constant difference or ratio; instead, each term equals the product of three consecutive integers taken in decreasing order, with the leading (largest) integer increasing by one from term to term: n × (n−1) × (n−2).

Application:

1. Write 24 as a product of three consecutive integers: 24 = 4×3×2.

2. Write 60 the same way: 60 = 5×4×3.

3. Write 120 the same way: 120 = 6×5×4.

4. Write 210 the same way: 210 = 7×6×5.

5. The leading integer rises by 1 at every step (4, 5, 6, 7, ...), so the next term uses 8 as the leading integer: 8×7×6.

6. 8×7×6 = 336.

Cross-check:

Look instead at the differences between consecutive terms: 60 - 24 = 36, 120 - 60 = 60, 210 - 120 = 90. Each of these equals 6 times a triangular number: 6×6, 6×10, 6×15 (the triangular numbers 6, 10, 15). The next triangular number in that sequence is 21, giving a difference of 6×21 = 126, so 210 + 126 = 336 - the same result as the product method above.