In this guide
Number series questions are a regular feature of bank PO prelims, usually as "find the missing term" or "find the wrong term". They look endless, but they are built from a small set of patterns. A candidate who checks those patterns in a fixed order finds most answers in 20 to 40 seconds. A candidate who stares at the numbers waiting for inspiration can lose two minutes on one question.
This guide gives you the routine, the common patterns with the reason each check works, six worked examples and a practice set.
Read the shape first
Before you calculate, look at how fast the numbers grow. That tells you where to start.
| How the series grows | Most likely pattern | Start with |
|---|---|---|
| Slowly, by similar steps | Additions: constant, squares, cubes or primes | First differences |
| Steps grow steadily | Differences that themselves form a pattern | Second differences |
| Roughly doubles or triples | Multiplication, often "× n + n" or "× 2 − 1" | Ratios of neighbours |
| Grows very fast | Rising multipliers or powers | Ratios, then squares and cubes |
| Falls fast | Division, often by falling numbers | Ratios the other way |
| Goes up and down | Two series mixed together | Alternate terms |
The checking routine
- First differences. Constant? Squares (1, 4, 9, 16…)? Cubes? Primes (2, 3, 5, 7, 11…)? A steady step?
- Second differences. If the first differences look random, take their differences.
- Multipliers. Divide each term by the one before. Look for ×2, ×3, rising multipliers (×1.5, ×2, ×2.5…), or "× n + n" and "× n − n".
- Squares and cubes. Terms close to n² or n³, plus or minus a constant.
- Alternate terms. Two series running at odd and even positions.
- Two readings agree. Some series fit two rules at once. That is fine, as long as both give the same next term.
Why the routine works: a series built by adding produces a pattern in the differences, and a series built by multiplying produces a pattern in the ratios. Checking both in order covers most question setters' building blocks.
Worked examples
Example 1 (square differences). 17, 26, 42, 67, 103, ?
- Differences: 9, 16, 25, 36. These are 3², 4², 5², 6².
- Next difference 7² = 49, so the answer is 103 + 49 = 152.
Example 2 (× n + n). 6, 7, 16, 51, 208, ?
- 6 × 1 + 1 = 7; 7 × 2 + 2 = 16; 16 × 3 + 3 = 51; 51 × 4 + 4 = 208.
- Next: 208 × 5 + 5 = 1,045.
- Tell-tale sign: the ratio of neighbours rises by about 1 each time.
Example 3 (rising multipliers). 8, 12, 24, 60, 180, ?
- Ratios: 1.5, 2, 2.5, 3. The multiplier rises by 0.5.
- Next: 180 × 3.5 = 630.
Example 4 (second differences). 12, 14, 20, 32, 52, ?
- First differences: 2, 6, 12, 20. No obvious pattern.
- Second differences: 4, 6, 8. Next second difference is 10, so the next first difference is 30.
- Answer: 52 + 30 = 82.
Example 5 (alternate series). 3, 20, 6, 17, 12, 14, 24, ?
- The series goes up and down, so split it.
- Odd positions: 3, 6, 12, 24 (× 2).
- Even positions: 20, 17, 14, ? (− 3).
- The missing term is in an even position: 14 − 3 = 11.
Example 6 (wrong term). 5, 9, 17, 33, 66, 129
- Try × 2 − 1: 5 → 9 → 17 → 33 → 65 → 129.
- Every term fits except the fifth. The wrong term is 66; it should be 65.
More patterns to recognise
| Pattern | Example | Rule | Next term |
|---|---|---|---|
| Cubes plus a constant | 3, 10, 29, 66, 127 | n³ + 2 | 218 |
| Prime differences | 11, 13, 16, 21, 28, 39 | + 2, + 3, + 5, + 7, + 11 | 52 |
| Doubling differences | 4, 6, 10, 18, 34 | Differences 2, 4, 8, 16; also × 2 − 2 | 66 |
| Falling divisors | 5,040, 720, 120, 24, 6 | ÷ 7, ÷ 6, ÷ 5, ÷ 4, ÷ 3 | 2 |
| Squares of a growing sequence | 1, 9, 36, 100, 225 | 1², 3², 6², 10², 15² | 441 (21²) |
The doubling-differences row is a good example of step 6: "differences double" and "× 2 − 2" both give 66, so you can answer with confidence.
Handling the wrong-term format
Wrong-term questions are harder than missing-term ones, because the error disturbs the pattern you are trying to find.
- Find the rule from the first three or four terms, which are usually correct.
- Generate the series yourself from the first term and compare term by term.
- If two terms seem wrong, your rule is wrong. A correctly built series has exactly one error.
- Check the last term. If your rule reaches it only after "correcting" a middle term, you have found the error.
Practice set
- 9, 10, 14, 23, 39, ?
- 4, 5, 12, 39, 160, ?
- 1, 9, 36, 100, 225, ?
- Find the wrong term: 7, 15, 31, 63, 128, 255
- 20, 24, 33, 49, 74, ?
- 11, 13, 16, 21, 28, 39, ?
- 5,040, 720, 120, 24, 6, ?
- Find the wrong term: 2, 8, 20, 44, 92, 184
Answers:
- 64. Differences 1, 4, 9, 16, so the next is 25: 39 + 25.
- 805. × 1 + 1, × 2 + 2, × 3 + 3, × 4 + 4, then 160 × 5 + 5.
- 441. Squares of 1, 3, 6, 10, 15 (gaps grow by one), so the next is 21².
- 128. The rule is × 2 + 1, which gives 127.
- 110. Differences 4, 9, 16, 25 (2² to 5²), so the next is 36: 74 + 36.
- 52. Differences are the primes 2, 3, 5, 7, 11, so the next is 13: 39 + 13.
- 2. Division by 7, 6, 5, 4, then 3: 6 ÷ 3.
- 184. The rule is × 2 + 4: 2 → 8 → 20 → 44 → 92 → 188. The last term should be 188.
What to do next
- Write the "read the shape" table from memory until you can use it without looking.
- Solve ten series a day for two weeks, timing each one; aim for under 40 seconds.
- Keep a list of every new pattern you meet in mocks.
- Pair this with simplification and approximation and quadratic comparisons for a complete prelims quick-types block.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the State Bank of India website .
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