Series questions give a short list of numbers or letters and ask for the next term. They look like a test of insight and are actually a test of method: there are perhaps ten common rules, and every question uses one of them or a combination.
The time cost matters as much as the accuracy. A candidate who stares at a sequence hoping to recognise it can burn two minutes on a question worth one mark. A candidate with a checklist either finds the rule in twenty seconds or moves on.
Apply this checklist in order, and stop as soon as the rule becomes obvious.
The checklist is ordered by the likelihood of each rule, not by elegance, so following it in order is the fastest path in expectation.
The rule families, with an example of each.
| Rule | Series | Next term |
|---|---|---|
| Constant difference | 7, 11, 15, 19 | 23 (add 4) |
| Growing difference | 2, 5, 10, 17, 26 | 37 (differences 3, 5, 7, 9, then 11) |
| Constant ratio | 3, 6, 12, 24, 48 | 96 (multiply by 2) |
| Increasing ratio | 1, 2, 6, 24, 120 | 720 (multiply by 2, 3, 4, 5, then 6) |
| Perfect squares | 1, 4, 9, 16, 25 | 36 (1², 2², 3² ...) |
| Primes | 2, 3, 5, 7, 11, 13 | 17 |
| Sum of previous two | 1, 1, 2, 3, 5, 8 | 13 |
| Alternating rules | 2, 10, 4, 20, 6 | 30 (odd positions add 2, even positions add 10) |
A series can be a known sequence with a twist. The squares 1, 4, 9, 16, 25 become 2, 5, 10, 17, 26 when each is increased by one — and that second series is exactly the "growing difference" row above, which is why writing differences finds it quickly. Similarly, the primes appear shifted by one or embedded in a longer pattern.
Descents behave the same way. 100, 96, 88, 76 has differences of −4, −8, −12, so the next difference is −16 and the next term is 60. A series with growing negative differences is not harder than one with growing positive differences; it is the same rule seen from below.
Letter series are number series in disguise. Write the position of each letter in the alphabet and the sequence becomes numeric. B, D, G, K, P is 2, 4, 7, 11, 16, with differences of 2, 3, 4, 5, so the next difference is 6 and the next letter is the 22nd, which is V.
Letters can also move through the alphabet in blocks. AZ, BY, CX pairs each forward-step with a backward-step: the first letter moves forward by one while the second moves backward by one. When you see pairs, check the two positions separately rather than as a rigid pair.
Odd-one-out questions use the same knowledge backwards. Find the property that all but one of the options share. 4, 9, 16, 25, 36 and 50: all are perfect squares except 50. State the property before you state the odd one, or you will pick an option that is merely unusual.
2, 5, 10, 17, 26, ?
Write the differences: 3, 5, 7, 9. They are not constant, but they increase by two each time, so the difference sequence is itself arithmetic with a constant second difference. The next difference is 11, and 26 + 11 is 37.
Now notice the alternative view: the terms are 1² + 1, 2² + 1, 3² + 1, 4² + 1, 5² + 1. That is 2, 5, 10, 17, 26 exactly. The two descriptions are the same series, and the second explains why the first difference grows by two each time — a square pattern always produces odd differences.
This is worth remembering as a shortcut rather than a coincidence. Whenever the differences are 3, 5, 7, 9 — consecutive odd numbers — the terms are squares with a constant added, and you can write the next term straight from the square formula without continuing the difference table.
What comes next: 100, 96, 88, 76, ?
What comes next: B, D, G, K, P, ?
Which is the odd one out: 4, 9, 16, 25, 36, 50?
Without looking back: what do you do when the differences between terms are not constant, and what is the first thing to write for a letter series?
The answer is: write the differences of the differences, testing for a constant second difference; and for a letter series, write each letter's alphabet position so it becomes a number series.
Fill in the blanks from memory:
A series whose terms increase by a constant amount is called , while one that multiplies by a constant is called . In the alphabet, the letter B has position .
Write one series for each rule family on this page: constant difference, growing difference, constant ratio, squares, primes, and sum of the previous two. Then swap them with a friend, or return to them the next day, and solve them using only the checklist.
3, 8, 15, 24, 35, ? The differences are 5, 7, 9, 11 — consecutive odd numbers starting from 5, so the next is 13 and the answer is 48. These are the squares minus one.
2, 6, 12, 20, 30, ? The differences are 4, 6, 8, 10, so the next is 12 and the answer is 42. These are consecutive products, or n² + n.
1, 3, 9, 27, ? The ratio is a constant 3, so the answer is 81. This is the geometric case, and it should be caught at step three of the checklist before any difference table is written.
Writing these yourself is what converts the checklist from a list you remember into a habit you apply. Notice too that the first two both look unfamiliar until the differences are written down, and the third does not need differences at all — recognising which of the five steps to run, and in what order, is most of the skill.
That completes the analytical question types. The drill that follows puts them together under a clock, because the analytical section rewards a fast, practised routine more than any other. After that, the course turns to the training programme that turns technique into a score: mock tests, the error log, targets and exam-day execution.