Number sequences: test differences before guessing

When a number sequence looks unfamiliar, write down the change between neighbouring values. Differences can reveal a pattern that is difficult to notice from the original numbers alone.

A first worked example

Consider 3, 7, 13, 21, followed by a missing value. The differences are 4, 6 and 8. If the intended rule continues increasing the difference by 2, the next difference is 10 and the next value is 31.

Now check the rule from the beginning: 3 plus 4 is 7; 7 plus 6 is 13; 13 plus 8 is 21. A rule that explains only the last step is weaker than one that consistently explains every supplied step.

Try a second layer when needed

For 2, 5, 10, 17, the first differences are 3, 5 and 7. Their differences are all 2. This points to a steadily increasing first difference, giving 26 as the next value under that rule.

Do not assume every question uses addition. A sequence such as 2, 6, 18, 54 suggests repeated multiplication by 3 instead. Another question might alternate two operations or interleave separate sequences. Test the simpler relationships before constructing complicated exceptions.

Keep ambiguity in mind

A finite list of numbers can often be extended in more than one mathematical way. A practice question needs enough context and suitable options to indicate an intended rule. Your task is to find a consistent rule supported by the supplied information, not to prove that no conceivable formula could produce another continuation.

Before selecting an answer, state your rule in a sentence and apply it to each known transition. After a mistake, decide whether you missed the pattern or made an arithmetic error. Those call for different practice: better rule testing in the first case, more careful calculation in the second. The examples here are original instructional examples, separate from your scored attempt.