Learn›Arithmetic ReasoningAR›Sequences
Number sequences
Four or five numbers and a blank. The test wants to know if you can find the rule, and it sets traps for anyone who guesses the rule from one gap. Here is the order in which to look.
Look at the gaps, not the numbers
A sequence follows a rule from one number to the next. The fastest way to see the rule is to write the change between each pair of neighbours under the gap between them. The changes tell you which kind of sequence it is.
Check at least two gaps
2 and 4 could come from adding 2 or from doubling: the next number is 6 in one case and 8 in the other. One gap proves nothing. Make sure the rule works for every gap you are given.
How to solve them
-
1
The same difference every time: keep adding it
If every gap shows the same difference, the sequence adds (or subtracts) that amount each time. This is called an arithmetic sequence.
Next step: MultiplyingAdding8, 15, 22, 29, ___
+7 each time → 36
Subtracting50, 44, 38, 32, ___
−6 each time → 26
-
2
The numbers grow faster and faster: try multiplying
If the differences keep getting bigger, divide each number by the one before it. If you get the same answer each time, the sequence multiplies by that amount. This is a geometric sequence.
Next step: Changing gapsMultiplying3, 12, 48, 192, ___
× 4 each time → 768
Halving400, 200, 100, 50, ___
÷ 2 each time → 25
-
3
When the differences change: look at the differences again
Sometimes the gaps are not equal but follow their own pattern. Write a second row: the differences of the differences.
Next step: Two patternsExample2, 5, 10, 17, 26, ___
- Differences: 3, 5, 7, 9.
- Those grow by 2 each time, so the next difference is 11.
- 26 + 11 = 37.
Answer 37
-
4
Two patterns in one: split it in two
If nothing works, look at every other number. Some sequences are two sequences woven together, or a pattern of two steps that repeats.
Next step: A far-away termExample1, 10, 3, 20, 5, 30, ___
- The 1st, 3rd and 5th numbers: 1, 3, 5 (add 2).
- The 2nd, 4th and 6th: 10, 20, 30 (add 10).
- The next one is in the first group: 5 + 2 = 7.
Answer 7
-
5
A far-away term without writing them all
For an adding sequence, the term in position n is the first term plus (n − 1) steps. It is n − 1, not n, because the first term has had no steps added yet.
term n = first + (n − 1) × step
Next step: The orderExample7, 11, 15, 19, … What is the 30th term?
- First term 7, step +4.
- 7 + (30 − 1) × 4 = 7 + 29 × 4 = 7 + 116 = 123.
Answer 123
-
6
The order to try things
Work through the same short list every time. Most sequences on the test fall to the first two checks.
1 · SubtractIs the difference the same each time?
Yes → add it again
2 · DivideIs the ratio the same each time?
Yes → multiply by it again
3 · Look deeperDo the differences have a pattern? Every other number?
Second differences, or split in two
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Assuming it adds
5, 15, 45, 135, ___
Wrong135 + 90 = 225
Right135 × 3 = 405
Using n instead of n − 1
The 30th term of 7, 11, 15, …
Wrong7 + 30 × 4 = 127
Right7 + 29 × 4 = 123
Trusting one gap
6, 12, 24, 48, ___
Wrongfrom 6 to 12 is +6 → 54
Right× 2 every time → 96
Losing the minus sign
50, 44, 38, 32, ___
Wrong32 + 6 = 38
Right32 − 6 = 26
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is the next number in the sequence 13, 21, 29, 37, ___?
Show the answer
B. Every gap is +8: 21 − 13 = 8, 29 − 21 = 8, 37 − 29 = 8. So 37 + 8 = 45. 41 adds only 4. The next number is 45.
What is the next number in the sequence 2, 10, 50, 250, ___?
Show the answer
D. The gaps grow fast (8, 40, 200), so divide: 10 ÷ 2 = 5, 50 ÷ 10 = 5, 250 ÷ 50 = 5. Each number is 5 times the one before: 250 × 5 = 1,250. 450 adds the last gap again. The next number is 1,250.
What is the next number in the sequence 1, 4, 9, 16, 25, ___?
Show the answer
C. Differences: 3, 5, 7, 9. They grow by 2, so the next one is 11, and 25 + 11 = 36. (They are the square numbers 1², 2², 3², …, and 6² = 36.) The next number is 36.
In the sequence 12, 17, 22, 27, …, what is the 40th term?
Show the answer
A. First term 12, step +5. The 40th term is 12 + (40 − 1) × 5 = 12 + 195 = 207. 212 multiplies by 40 instead of 39. The 40th term is 207.
What is the next number in the sequence 3, 8, 6, 11, 9, 14, ___?
Show the answer
B. The steps go +5, −2, +5, −2, +5. The next step is −2: 14 − 2 = 12. 19 keeps adding 5. The next number is 12.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished number sequences.
Practice score: not taken yet Take the practice
Next topic: Charts & graphsSee this topic move.
In the course, every topic is drawn step by step on a whiteboard while a voice explains it, with timed practice tests and a plan for your test date. The first lessons of every module are free.
