Most wrong answers in a number series test come from the same habit: the first two numbers suggest a rule, the mind locks onto it, and the rest of the row never gets checked. A short routine fixes that.
Step 1: Look at the shape before you calculate
Spend a second on how the row moves. Does it climb slowly and evenly? Does it explode? Does it go up, down, up, down? Each shape points to a different family.
- Slow, steady climb: probably adding, or gaps that grow slowly.
- Fast climb, each term several times the last: probably multiplying, or squares and cubes.
- A zigzag: probably two series woven together.
- Falling: subtracting, dividing, or shrinking gaps.
Step 2: Write the gaps under the numbers
Take each number away from the one after it and write the result in a second row. If the second row is constant, you are done: it is a plain staircase.
100, 94, 85, 73, 58, ? gives gaps of −6, −9, −12, −15. The gaps are not constant but they are regular: each is 3 more negative than the last. The next gap is −18, so the term is 58 − 18 = 40.
Step 3: If the gaps move, take the gaps of the gaps
1, 2, 4, 7, 11, 16, ? has gaps 1, 2, 3, 4, 5. The gaps climb by exactly 1, which is a constant second row, so the next gap is 6 and the term is 22.
Gaps can also be a recognisable list instead of a staircase. In 3, 4, 8, 17, 33, 58 the gaps are 1, 4, 9, 16, 25, the square numbers, so the next gap is 36 and the term is 94.
Step 4: Try the ratios, then try splitting the row
If the gaps are huge and unruly but each number is a clean multiple of the last, divide instead. 2, 6, 18, 54: each term is three times the one before.
If neither works and the row zigzags, separate the odd positions from the even ones and solve each list on its own. 3, 50, 6, 45, 9, 40, 12, 35 splits into 3, 6, 9, 12 (adds 3) and 50, 45, 40, 35 (takes away 5). The ninth term is in the first list, so it is 15.
Step 5: Check the rule against every term
This is the step that people skip, and it is where the marks are. Run your rule from the first term all the way to the last number shown. If it reproduces them all, use it.
Take 4, 7, 13, 25, 49, ? The gaps are 3, 6, 12, 24, which double each time, so the next gap is 48 and the term is 97. Now check a second way: double each term and take away 1. 4 → 7, 7 → 13, 13 → 25, 25 → 49, and 49 → 97. Two routes, one answer, and every shown number accounted for.
Why the simplest rule wins
Any finite row of numbers can be extended in many ways if you allow rules that are complicated enough. A series question is not a trick question about that fact. It asks for the simple rule that explains every number shown, which is exactly what a person on the other side wrote.
Psychologists call this task numerical rule induction (Holzman, Pellegrino and Glaser, 1982), and the finding that matters here is practical: a rule has to be tested against all the evidence, not accepted on the first fit. So prefer the shortest rule that fits all the terms. If your rule needs a special case or a fix to cover one number, it is probably the wrong rule.
When you are stuck
Move on and come back; the clock matters more than any single item. When you return, try the chained-operations idea (double and add, then double and take away) and the sum-of-previous-terms idea, which are the two that the gap method hides best. You can rehearse the whole routine on the number series practice page or in the JobCannon Number Series test.