Find the missing term in the following series: 0, 6, 24, 60, 120, ?, 336,…

Concept

A polynomial number series can be tested by expressing its nth term with a simple identity. The product of three consecutive integers centred at n is (n − 1)n(n + 1) = n3 − n.

After identifying a candidate rule, check it against several known positions and the following term; one match alone is not enough.

Application

1. Number the terms from n = 1 onward.

2. For n = 1, 2, 3, 4 and 5, the rule n3 − n gives 0, 6, 24, 60 and 120 respectively.

3. At the sixth position, (6 − 1) × 6 × (6 + 1) = 5 × 6 × 7 = 210.

4. At the seventh position, (7 − 1) × 7 × (7 + 1) = 6 × 7 × 8 = 336, matching the next given term.

Cross-check

The successive differences are 6, 18, 36, 60, 90 and 126. Their second differences are 12, 18, 24, 30 and 36, increasing by 6 each time; therefore 120 + 90 = 210 and 210 + 126 = 336.

Result

Therefore, the missing term is 210.