Growing Gaps
What comes next? 2, 6, 12, 20, 30, ?
Look at how far apart the numbers are.
42
The gaps grow by two each time: +4, +6, +8, +10, then +12. 30 + 12 = 42.
How to solve it
Step one: look at the gaps between neighbouring numbers instead of the numbers themselves. From 2 to 6 is 4, from 6 to 12 is 6, from 12 to 20 is 8, and from 20 to 30 is 10.
Step two: spot the pattern in the gaps. They are 4, 6, 8, 10, growing by 2 each time, so the next gap is 12.
Step three: add it on. 30 + 12 = 42.
A second way to confirm it: each term is a number times the next one. 1 times 2 is 2, 2 times 3 is 6, 3 times 4 is 12, 4 times 5 is 20, 5 times 6 is 30, so the next is 6 times 7, which is 42. Two different methods giving the same result is a strong sign the answer is right.
The usual wrong answer is 40, from adding the last gap of 10 again as if the sequence went up by a fixed amount. Another slip is 44, from making the gap grow by 4 instead of 2, so 30 + 14. Always write the gaps down in a row; the rule usually shows itself there.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
