Take the differences between consecutive terms before anything else. Most number sequences on IQ tests are built on a rule that becomes obvious one level down: if the differences are constant the sequence is arithmetic, and if the differences themselves form a pattern you have found the rule. When differences fail, check ratios, then look for two interleaved sequences, then check whether each term is built from the ones before it.
The order to check things in
Working through the possibilities in a fixed order beats staring at the numbers, because it turns an open-ended search into four quick tests.
First, differences. Subtract each term from the next. Constant difference means arithmetic. If the differences form their own pattern, take differences again; two levels is enough for almost every test item.
Second, ratios. Divide each term by the previous one. A constant ratio means geometric. Watch for ratios that change in a pattern, such as multiply by 2 then 3 then 4.
Third, interleaving. Read alternate terms as two separate sequences. Series like 2, 10, 4, 20, 6, 30 make no sense read straight and resolve immediately when split.
Fourth, self-reference. Check whether each term comes from combining previous terms, as in Fibonacci where every term is the sum of the two before it.
A worked example
Take 2, 6, 12, 20, 30, and find the next term.
The differences are 4, 6, 8, 10. Constant? No. But they form a pattern of their own, increasing by 2 each time. Taking differences a second level down gives 2, 2, 2, which confirms it.
So the next difference is 12, and the next term is 30 + 12 = 42. This item is in our general test bank and it is a good example of why the difference check comes first: the original numbers look irregular and the structure appears immediately one level down.
A second example: 3, 7, 13, 21, 31. Differences are 4, 6, 8, 10, the same shape. Next difference is 12, so the answer is 43. Recognising a rule you have already met is much of what practice buys you.
The patterns that appear most often
| Type | Rule | Example |
|---|---|---|
| Arithmetic | Add a constant | 3, 7, 11, 15 |
| Second-order | Difference grows steadily | 2, 6, 12, 20, 30 |
| Geometric | Multiply by a constant | 3, 9, 27, 81 |
| Squares and cubes | Terms are n squared or n cubed | 1, 4, 9, 16, 25 |
| Primes | Consecutive prime numbers | 2, 3, 5, 7, 11 |
| Fibonacci-style | Each term is the sum of the previous two | 1, 1, 2, 3, 5, 8 |
| Interleaved | Two sequences alternating | 2, 10, 4, 20, 6, 30 |
| Alternating operation | Operations alternate | 2, 4, 3, 6, 5, 10 |
Knowing squares to 15 and cubes to 6 by sight is worth the ten minutes it takes. A surprising number of sequences are built on them, sometimes offset by a constant, and recognising 26 as 25 plus one saves the whole search.
Traps that catch people out
More than one rule fits. Any finite sequence can be continued by infinitely many rules, so test writers rely on the simplest rule being intended. If you find a rule requiring three operations and the options include an answer from a simpler one, the simpler one is what was meant.
Stopping at the first level. Differences that are not constant get abandoned too early. Take differences of the differences before moving on; second-order sequences are extremely common and one more subtraction resolves them.
Negative and fractional steps. Sequences that descend or cross zero read as chaotic when scanned. Subtract carefully and keep the signs.
Ignoring the options. On multiple choice, the answers constrain the problem. If three options are close together and one is far away, the far one is usually there for a rule you have not considered, and checking why it might be right sometimes reveals the actual pattern.
Logical Reasoning Test
25 questions, 28 minutes, scored as soon as you finish.
Take the Logical Reasoning TestQuestions people also ask
Is there always one correct answer to a number sequence?
Mathematically, no. Infinitely many rules can continue any finite sequence. Test items work on the convention that the simplest rule producing the given terms is the intended one, which is why well-written items avoid sequences where two simple rules both fit.
Do number sequences measure mathematical ability?
Only slightly. They use arithmetic as the medium but they measure pattern recognition, and the arithmetic involved rarely goes beyond addition and multiplication. People who dislike maths often do well on them once the format stops being intimidating.
Keep reading
- Matrix Reasoning Questions ExplainedThe rules behind the grids, and a method for solving them fast.
- How to Prepare for an IQ TestFormat familiarity, sleep, and the tactics that do nothing.
- IQ Percentile Chart: What Your Score Actually RanksThe full table from 70 to 145, and how percentiles are derived.