Number Series is one of the most dependable scoring topics in the Reasoning section of SSC, Railway, Banking, Police and State PSC exams. A series of numbers follows a hidden rule, and you must find the next or missing number. The rule is rarely complicated, but you will only spot it quickly if you know the common pattern families and check them in the right order.
Want a full 20-question MCQ drill? Try our Number Series 20 MCQs with solutions (Hindi) after reading this guide.
1. What Is a Number Series?
A number series (or number sequence) is a list of numbers arranged according to a fixed rule. The rule may use one operation, such as adding 5 each time, or a combination, such as multiplying by 2 and then subtracting 1. Questions usually appear in two forms: find the next term, or find the wrong or missing term somewhere inside the series.
The same few rules appear again and again, so your job is not to invent a rule from scratch. Your job is to test known rules quickly until one fits every term.
2. The Nine Pattern Types You Must Know
Type 1: Constant difference (Arithmetic)
The same number is added or subtracted every time. Example: 10, 17, 24, 31 (+7). Spot it by subtracting neighbouring terms and getting the same value.
Type 2: Constant ratio (Geometric)
Each term is multiplied or divided by the same number. Example: 2, 6, 18, 54 (×3). Spot it when terms grow or shrink very fast and the differences themselves keep growing.
Type 3: Increasing or decreasing differences
The gap between terms changes by a regular amount. Example: 1, 2, 4, 7, 11 (differences +1, +2, +3, +4). If the first differences are not equal, always write down the differences and examine them.
Type 4: Squares and cubes
The terms are perfect squares or cubes, such as 1, 4, 9, 16 or 8, 27, 64, 125. Learn the tables at the end of this section so you can recognise them instantly.
Type 5: Squares or cubes with a small adjustment
Terms follow n² + 1, n² − 1, n² + n, n³ + n or n³ − 1. Example: 2, 5, 10, 17, 26 is n² + 1. If the terms look close to squares or cubes but are slightly off, try this type.
Type 6: Multiply and then add or subtract
Each term is multiplied by a number and then a constant is added or subtracted. Example: 2, 4, 10, 28, 82 (×3 − 2). This is common in moderate-level questions.
Type 7: Alternating series
Two different series are mixed together. Terms in odd positions follow one rule, and terms in even positions follow another. Example: 2, 5, 4, 10, 8, 20 has 2, 4, 8 in the odd places and 5, 10, 20 in the even places. Suspect this when the series goes up and down, or when no single rule fits.
Type 8: Prime numbers and Fibonacci-type series
Prime series: 2, 3, 5, 7, 11, 13. Fibonacci-type series: each term is the sum of the previous two, as in 1, 1, 2, 3, 5, 8. If every term is roughly the sum of the two before it, test this rule.
Type 9: Second-order differences
The differences themselves do not form a simple pattern, but the differences of the differences do. Example: 1, 4, 10, 20, 35 has first differences 3, 6, 10, 15 and second differences 3, 4, 5.
3. Reference Tables to Memorise
Knowing these values saves several seconds per question, which adds up over a full paper.
| Number | Square |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| Number | Cube |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
4. A Reliable Step-by-Step Method
- Look at the size of the numbers. Slow growth suggests addition or squares. Fast growth suggests multiplication or cubes.
- Write the differences. If they are equal, you are done. If they change regularly, use Type 3 or Type 9.
- Check the ratio. Divide each term by the one before it. A constant answer means Type 2.
- Compare with squares and cubes. Terms near a perfect square or cube point to Type 4 or Type 5.
- Try multiply-and-adjust. If the ratio is close to constant but not exact, test Type 6.
- Split the series. If nothing fits, separate odd-position and even-position terms.
- Verify. Apply the rule to every term before you mark the answer.
5. Solved Examples
The difference is +7 each time, so the next number is 33 + 7 = 40.
Each term is multiplied by 4. The next number is 320 × 4 = 1280.
The differences are 1, 3, 5, 7, so the next difference is 9. The next number is 20 + 9 = 29.
Each term is n³ − 1: 1³−1, 2³−1, 3³−1, 4³−1, 5³−1. The next is 6³ − 1 = 216 − 1 = 215.
Each term is the previous term × 2 + 1. The next is 79 × 2 + 1 = 159.
Odd positions: 2, 4, 8, 16 (×2). Even positions: 5, 10, 20 (×2). The next term sits in an even position, so it is 20 × 2 = 40.
Each term is the sum of the previous two: 2+3=5, 3+5=8, 5+8=13. The next is 8 + 13 = 21.
The series is multiplied by 2 each time, so the missing term is 10 × 2 = 20. Check: 20 × 2 = 40.
6. Common Mistakes to Avoid
- Deciding the rule after checking only the first two terms. Always verify across the whole series.
- Forgetting to check the differences of the differences when the first differences are not equal.
- Missing alternating series because you only searched for one rule.
- Calculation slips in squares and cubes. Memorise the tables above.
- Spending too long on one question. If no pattern appears within 45 seconds, mark it and move on.
7. Concept Check: 5-Question Quiz
Choose an answer for each question, then press Submit. Green means correct, red means incorrect and yellow means unanswered. Explanations are listed under the quiz.
Show explanations
- 45. Add 9 each time (the table of 9): 36 + 9 = 45.
- 22. Differences are 2, 3, 4, 5, so the next difference is 6: 16 + 6 = 22.
- 1. Divide by 5 each time: 5 ÷ 5 = 1.
- 91. These are sums of squares: 1, 1+4, 1+4+9, 1+4+9+16, 1+4+9+16+25. Add 6² = 36 to 55 to get 91.
- 38. The rule alternates between ×2 and +2: 3×2=6, 6+2=8, 8×2=16, 16+2=18, 18×2=36, 36+2=38.
8. Frequently Asked Questions
Is Number Series important for competitive exams?
Number Series appears in the reasoning or quantitative aptitude section of many competitive exams, though the exact weightage depends on the exam and year. Check the latest syllabus and previous papers of your target exam.
How do I solve Number Series questions quickly?
Follow a fixed order: differences, ratio, squares and cubes, multiply-and-adjust, then alternating terms. Practising this order makes pattern spotting almost automatic.
What should I do when no pattern fits?
Split the series into odd and even positions, and look at second-order differences. If neither works, move on and return later if time permits.
How many questions should I practise every day?
Ten to fifteen questions a day, with a review of every mistake, is usually enough to build speed over a few weeks. Quality of review matters more than quantity.
Where can I find more practice questions?
After you are comfortable with the patterns here, solve our 20 Number Series MCQs with solutions and time yourself.
Conclusion
Number Series becomes easy once you know the nine pattern types, memorise squares and cubes and test patterns in a fixed order. Read the solved examples again, take the quiz twice, and then move on to timed MCQ practice.
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