Two questions open most number series. What do I get if I subtract neighbours? What do I get if I divide them? Add a third move, subtracting again, and you can read the family a series belongs to from the numbers alone.
Question one: what are the gaps?
Subtract each term from the next. If every gap is identical, the series adds (or subtracts) a fixed step. 17, 23, 29, 35, 41 has a gap of 6 every time, so the next term is 47.
Then look at the second row. If it is constant, you have your rule. If it is not, that row is not a dead end: it is the next thing to read.
The second row: gaps of the gaps
When the gaps are not constant, write them and take their gaps. 1, 2, 4, 7, 11, 16 has gaps 1, 2, 3, 4, 5, and those gaps differ by a steady 1. The next gap is 6, so the next term is 22.
2, 5, 11, 20, 32 has gaps 3, 6, 9, 12, which differ by a steady 3. The next gap is 15, so the term is 47.
It works downward too. 100, 94, 85, 73, 58 has gaps of −6, −9, −12, −15, each 3 lower than the last. The next gap is −18, giving 40.
When the gaps are a recognisable list
Sometimes the gaps are neither constant nor in steady steps, but they are a list you know. In 3, 4, 8, 17, 33, 58 the gaps are 1, 4, 9, 16, 25. Those are the squares, so the next gap is 36 and the term is 94.
And in 1, 3, 7, 15, 31 the gaps are 2, 4, 8, 16. The gaps double, so the next is 32 and the term is 63. The same series can be read as double the last and add 1; the two readings agree, which is a reassuring sign.
The fingerprint of square and cube numbers
Squares leave a recognisable trace. 4, 9, 16, 25 has gaps 5, 7, 9 and a constant second row of 2. The step from n² to (n+1)² is always 2n+1, which is why.
Cubes need one more row. 1, 8, 27, 64, 125 has gaps 7, 19, 37, 61, then second differences 12, 18, 24, then a constant third row of 6. When a third row finally goes flat, you are looking at cubes.
Question two: what are the ratios?
Divide each term by the one before. 2, 6, 18, 54, 162 gives 3 every time, so the next term is 486. 3, 12, 48, 192 gives 4 and the next term is 768.
Reach for the ratio first when the terms grow fast. If each number is at least double the last, gaps are rarely the answer.
A quick decision list
- Gaps constant: add or subtract that step.
- Gaps change by a constant: add the next gap, which is the last gap plus the change.
- Gaps are squares, or double each time: use that list for the next gap.
- Ratios constant: multiply by that ratio.
- None of the above: split into odd and even positions, then try a pair of operations or a sum of earlier terms.
Practise the decision on real rows in the JobCannon Number Series test. Its results are broken out by kind of rule, so you can see whether growing gaps or steady steps are your weaker side. For the wider skill of working with ratios and percentages in tables, the JobCannon Numerical Reasoning test is the companion.