About Number Series
A number series is a rule applied repeatedly, with the rule hidden. The task is not to guess but to test the simplest rules first and stop as soon as one of them fits every term you can see. Candidates lose marks here mainly by committing to a rule too early and then forcing the final term to agree with it.
What you need to understand
- Write the differences between consecutive terms underneath the series. Most rules announce themselves there.
- If the differences are constant, the series is arithmetic. If the differences themselves change by a constant amount, the series is quadratic.
- If the ratios between consecutive terms are constant, the series is geometric.
- If the differences are perfect squares, cubes, primes or consecutive odd numbers, the rule lies in the differences rather than in the terms.
- If the terms grow much faster than a constant ratio would allow, look for a rule that multiplies and then adds, or for squares and cubes.
- Before choosing an answer, check whether a simpler rule also fits every given term. If two rules fit, the simpler one is intended.
How to work through these questions
- Write the series, then write the gaps between consecutive terms directly below it.
- If the gaps are equal, add the same gap again and stop.
- If the gaps are not equal, write the gaps between the gaps. A constant second difference means the first differences are increasing steadily.
- If the gaps are not arithmetic, check the ratios between consecutive terms, then check whether the terms sit near perfect squares or cubes.
- If none of these work, test a rule of the form multiply by m and add c, and test alternating patterns where two operations take turns.
- Finally, verify the rule reproduces every given term, not just the last two, before you commit to an answer.
Mistakes that cost marks
- Assuming the series is arithmetic because the first three terms happen to be evenly spaced.
- Looking only at the last two terms and inventing a rule that explains those but not the rest of the series.
- Treating a geometric series as arithmetic, or the reverse, when the differences look small at the start.
- Ignoring a rule that alternates between two operations, such as add, double, add, double.
- Choosing an answer that fits an equally valid but more complicated rule than the intended one.
Worked example
Find the next term of the series: 3, 6, 11, 18, 27, ?
- Write the gaps underneath the terms: 3, 5, 7, 9.
- The gaps are not equal, so the series is not arithmetic. Write the gaps between the gaps: 2, 2, 2 — a constant second difference.
- The gaps are therefore the consecutive odd numbers 3, 5, 7, 9 and the next gap is 11.
- Add it to the last term: 27 + 11 = 38.
- Check the rule against every term: 3, 3+3=6, 6+5=11, 11+7=18, 18+9=27, 27+11=38. All terms agree.
Answer: 38
Practice questions with answers
A few Number Series questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Work out the next term of the series: 1, 2, 4, 8, 16, 32, ?
Answer: Option B — with explanation
Every term is 2 times the term before it.
Multiply the previous term by 2: 32 x 2 = 64.
Common mistakes
- continued it as a difference series: 96
- applied the common ratio one time too many: 128
- an arithmetic slip in the final step: 61
Question 2
What should replace the question mark (?) in the following series: 1, 2, 4, 8, ?, 32, 64
Answer: Option B — with explanation
Every term is 2 times the term before it.
Multiply the previous term by 2: 8 x 2 = 16.
Common mistakes
- continued it as a difference series: 96
- applied the common ratio one time too many: 128
- an arithmetic slip in the final step: 13
Question 3
Which number comes next in the series: 2, 7, 17, 37, 77, 157, ?
Answer: Option D — with explanation
Each term is 2 times the previous term, plus 3.
Multiply the previous term by 2 and add 3: 157 x 2 + 3 = 317.
Common mistakes
- added the constant instead of multiplying: 160
- forgot to add the constant in the last step: 314
- added the constant twice in the last step: 320
Question 4
What should replace the question mark (?) in the following series: 2, 15, ?, 171, 522, 1575, 4734
-
A
1584
-
B
54
-
C
4743
-
D
4725
Answer: Option B — with explanation
Each term is 3 times the previous term, plus 9.
Multiply the previous term by 3 and add 9: 15 x 3 + 9 = 54.
Common mistakes
- added the constant instead of multiplying: 1584
- forgot to add the constant in the last step: 4725
- added the constant twice in the last step: 4743
Question 5
Find the next number in the series: 2, 5, 11, 23, 47, 95, ?
Answer: Option C — with explanation
Each term is 2 times the previous term, plus 1.
Multiply the previous term by 2 and add 1: 95 x 2 + 1 = 191.
Common mistakes
- forgot to add the constant in the last step: 190
- added the constant twice in the last step: 192
- added the constant instead of multiplying: 96
Frequently asked questions
What is the fastest way to solve a number series question?
Write the differences between consecutive terms. Constant differences mean an arithmetic series, and differences that increase steadily mean a quadratic one. Those two checks solve the majority of questions.
What if the differences do not show any pattern?
Check the ratios between consecutive terms for a geometric series, then check whether the terms are close to perfect squares or cubes. If the terms grow very fast, test a rule that multiplies by one number and adds another.
How do I handle a series where a number is missing in the middle?
Work out the rule using the terms on both sides of the gap, then check that the rule reproduces every visible term. Using terms after the gap as well as before it is what prevents a wrong answer.
Can a series have more than one valid answer?
Mathematically, yes — several rules can sometimes fit the visible terms. In an exam the intended answer is always the simplest rule that fits every term, which is why checking a simpler alternative before answering is worth the few seconds.
Are letter series solved the same way?
Yes. Convert the letters into their positions in the alphabet, 1 to 26, and then treat the result as a number series. The same difference and ratio checks apply.
