Which number comes next in the following sequence? 2, 6, 12, 20, 30, 42, 56,…

Concept

A number-series pattern can be tested by expressing the term at position n as a rule involving n.

If an = n(n + 1), the terms are products of consecutive integers; equivalently, the successive differences are consecutive even numbers.

Application

1. For n = 1, 1 × 2 = 2; for n = 2, 2 × 3 = 6; and the same rule gives the displayed terms.

2. For n = 7, 7 × 8 = 56, which agrees with the last displayed term.

3. The next position is n = 8, so 8 × 9 = 72.

Cross-check

The differences are 4, 6, 8, 10, 12, and 14. The next difference is 16, and 56 + 16 = 72.

Result

Therefore, the next term is 72.