The 2, 6, 12, 20 Sequence

What number comes next?
2, 6, 12, 20, 30, ?
💡 Hint

The gaps are 4, 6, 8, 10…

Reveal answer

42

Method 1: the differences between terms are 4, 6, 8 and 10, growing by 2 each time. The next gap is 12, so 30 + 12 = 42. Method 2: each term is a number multiplied by the next one: 1×2 = 2, 2×3 = 6, 3×4 = 12, 4×5 = 20, 5×6 = 30, so the next term is 6×7 = 42. Both methods give the same answer, which is a good sign you have found the real rule.

Did you know?These are called pronic or oblong numbers, because each one can be drawn as a rectangle of dots that is one dot longer than it is wide. They are also exactly twice the triangular numbers (1, 3, 6, 10, 15, 21). When a sequence puzzle stumps you, checking the differences, then the differences of the differences, cracks a surprising number of them.

About This Puzzle

Number sequences are a staple of brain training because there is always a rule hiding behind the numbers. This one has a rule that you can spot in two different ways.

How to Think About It

Start by looking at the gaps between the numbers instead of the numbers themselves. Write the differences under the sequence. If the gaps follow a simple pattern, you can extend them by one step and add.

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