Q1 What comes next: 3, 6, 11, 18, 27, …?
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38.
Differences: 3, 5, 7, 9 - odd numbers increasing by 2. The next difference is 11, so the next term is 27 + 11 = 38. (The series is n² + 2 for n = 1, 2, 3, …)
Number Series
What it is
A number series question has exactly one intended rule, and there is a standard order for hunting it. Differences first; if those are not constant, differences of the differences; then ratios; then squares and cubes; then alternating series.
Going through that list takes about fifteen seconds and finds the rule in most questions. Staring at the numbers hoping to see it does not.
What you need to know
- Take first differences. Constant means arithmetic; the rule is add d.
- If the first differences are not constant, take their differences. Constant there means a quadratic - often n² with an offset.
- Take ratios. Constant means geometric; the rule is multiply by r.
- Compare against 1, 4, 9, 16, 25 and 1, 8, 27, 64, 125 - squares and cubes, often shifted by one.
- If nothing fits, try reading alternate terms as two interleaved series.
Worked examples
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q2 What comes next: 2, 6, 12, 20, 30, …?
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42.
Differences: 4, 6, 8, 10, rising by 2, so the next is 12 and the term is 42. The same series is n(n + 1): 1×2, 2×3, 3×4, 4×5, 5×6, 6×7.
Q3 Find the odd one out: 8, 27, 64, 100, 125, 216, 343.
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100.
Every other term is a perfect cube - 2³, 3³, 4³, 5³, 6³, 7³. 100 is a perfect square but not a cube, which is exactly why it was chosen: it looks at home among them.
Where people lose marks
- Finding a rule that fits the first three terms and not the rest. Check your rule against every given term.
- Missing an interleaved pair of series when the terms jump up and down.
- Forcing an arithmetic rule onto a series that is multiplicative.