IQ explained · 6 min read
How to solve number series problems
A method for number series questions: differences, ratios, alternation, recursion, squares, cubes and spotting ambiguity.
In short
Number series questions ask what comes next. Almost all of them are built from five mechanisms: a constant difference, a constant ratio, differences that themselves change by a rule, alternation between two interleaved rules, or recursion where each term depends on the previous ones. Writing the differences between adjacent terms usually exposes the mechanism in one step. If two mechanisms fit, the question is ambiguous and the simplest rule is the best-supported answer.
Try it on yourself: Numerical reasoning questions with answers · 12 min · free, with the full result.
Practising for a timed test? Numerical reasoning test practice
Untimed and new every time: Untimed puzzles, new every time
New series with a nudge, the rule, then the working: Number sequence test with hints
Letters as well as numbers: letter and number patterns
Step one: write the differences
Under the series, write the difference between each pair of neighbours. For 7, 11, 15, 19 the differences are 4, 4, 4 and the next term is 23. For 2, 3, 5, 9, 17 the differences are 1, 2, 4, 8, which double, so the next difference is 16 and the next term is 33. If the differences are not constant, write the differences of the differences; a constant second difference means a quadratic rule such as squares.
Step two: test ratios
If differences grow quickly, divide neighbours instead. For 3, 6, 12, 24 each term is double the last, so the next is 48. For 81, 27, 9, 3 each term is a third of the last, so the next is 1. Ratios also expose mixed rules such as "double, then subtract two": 5, 10, 8, 16, 14, 28 alternates ×2 and −2, so the next term is 26.
Step three: look for alternation and recursion
Some series interleave two simple rules. In 10, 8, 9, 7, 8 the pattern is −2, +1, −2, +1, so the next term is 6. Others are recursive: in 1, 1, 2, 3, 5, 8 each term is the sum of the previous two. A quick test for recursion is to add or multiply the two previous terms and see whether the result matches the next one.
Step four: check the famous sequences
- Squares: 1, 4, 9, 16, 25, 36.
- Cubes: 1, 8, 27, 64, 125.
- Triangular numbers: 1, 3, 6, 10, 15, 21.
- Factorials: 1, 2, 6, 24, 120.
- Primes: 2, 3, 5, 7, 11, 13.
These appear often because they are easy to write and hard to guess without knowing them. Recognising them saves the arithmetic.
When a series has more than one answer
Any finite series can be continued by more than one rule; a well-written question makes one rule clearly simplest. If you find two rules that fit, choose the one with fewer steps, and note that in real tests the ambiguity is the question writer's error, not yours. Our numerical and pattern tests name the intended rule for every question in the answer review, so you can see which rule it follows.
Three series, worked with hints
Three fixed series, each solved with the same three hints the practice gives: where to look, the rule, then the worked solution. Write the gaps between the terms first, then open the hints in order and compare.
Learn the rules, try a puzzle and use a hint when you need one.
Your answers are never sent to us. We count how many hints are opened, never which answer you chose; on test pages, masked session replay may record your clicks (privacy section 2.6).
Example 1: Number series
What number comes next?
2610141822?
Hint 1: where to look
Choose a pair of neighbours near the start and one near the end. Work out how far each pair moves.
Hint 2: the rule
Subtract neighbours to find the difference: +4, +4. If the difference is constant, carry it on from the final term.
Hint 3: worked solution
- Subtract neighbours: you get +4, +4 at the start and the same gap at the end of 18, 22.
- Carry the series on by that gap once more, which gives 22 + 4 = 26.
Now find it among the options.
Answer: why each option fits or fails
Answer: B. 26.
- A 25Option A is close, yet one too few. Recheck the final calculation.
- B 26✓ AnswerThe answer is option B, which is 26.
- C 27Option C is one too many against the true next term, so the final step was slightly off.
- D 30Option D ends one step too far: count the steps again.
Example 2: Number series
What number comes next?
192964824126?
Hint 1: where to look
Read the 6 terms from the left. Say how each relates to the one before. Is it always the same?
Hint 2: the rule
Test a division: 192 → 96 gives a factor of 2. Does the next pair give the same factor?
Hint 3: worked solution
- Check the divisor on the last pair: it is still 2.
- Divide the end of 12, 6 once more: 6 ÷ 2 = 3.
Now find it among the options.
Answer: why each option fits or fails
Answer: A. 3.
- A 3✓ AnswerThe correct one is option A: 3.
- B 0Option B repeats the last gap, but this series does not keep one fixed gap.
- C 6Option C matches a local pattern near the end, but the same rule fails at the start.
- D -24Option D continues every other term at a fixed gap; the rule for every term gives another number.
Example 3: Number series
What number comes next?
17341937214023?
Hint 1: where to look
Look at the first, third and fifth terms together. Do they make a pattern of their own?
Hint 2: the rule
Split the list by position into two series. The odd positions begin 17, 19, the even positions 34, 37. Find each step.
Hint 3: worked solution
- Read the list as two runs.
- The end, 40, 23, shows both at work, and the next term sits in the even-position series (34, 37, 40).
- Give that run its own step: 40 + 3 = 43.
Now find it among the options.
Answer: why each option fits or fails
Answer: C. 43.
- A 6Option A repeats the last gap, but this series does not keep one fixed gap.
- B 44Option B is one too many against the true next term, so the final step was slightly off.
- C 43✓ AnswerOption C is correct: 43.
- D -30Option D fits only the last few terms; across the whole list two series take turns.
When the three make sense, try new series with the same hints: Number sequence test with hints.