The pattern families
| Family | Clue | Example |
|---|---|---|
| Constant difference | Equal gaps | 5, 9, 13, 17, 21 |
| Changing difference | Gaps form a pattern | 3, 4, 6, 9, 13 (gaps 1, 2, 3, 4) |
| Multiplying | Ratio constant or changing | 2, 6, 18, 54 |
| Squares and cubes | Terms near squares or cubes | 1, 4, 9, 16, 25 |
| Alternating series | Two interleaved patterns | 2, 5, 4, 8, 6, 11 |
| Add then multiply | Operation repeats in a cycle | 3, 7, 15, 31 (× 2 + 1) |
| Prime or other special numbers | Primes, Fibonacci | 2, 3, 5, 7, 11 |
A method for spotting the pattern
- Take differences between consecutive terms. If they are constant or follow a pattern, you are done.
- Take ratios if the terms grow fast.
- Check for squares or cubes (or one more or less than them).
- Look at alternate terms if the series seems erratic.
- Try a combined rule: × a ± b.
- Test the rule on every term before answering.
Worked examples
Example 1. 3, 7, 15, 31, ? The differences are 4, 8, 16, so the next is 31 + 32 = 63. Also each term is double the previous plus 1: 31 × 2 + 1 = 63.
Example 2. 2, 6, 12, 20, 30, ? The differences are 4, 6, 8, 10; next difference 12; term 42. (The terms are n(n + 1).)
Example 3. 1, 8, 27, 64, ? Cubes: next 5³ = 125.
Example 4 (alternating). 2, 5, 4, 8, 6, 11, 8, ? Odd positions: 2, 4, 6, 8 (+2). Even positions: 5, 8, 11 (+3), so the next in that line is 14, and the next term to find is at position 8: 14.
Example 5. 5, 11, 23, 47, ? Each term is double the previous plus 1: 47 × 2 + 1 = 95.
Example 6 (squares ± 1). 2, 5, 10, 17, 26, ? These are n² + 1: next 6² + 1 = 37.
Example 7 (ratio). 4, 12, 36, 108, ? Multiply by 3: 324.
A harder example, step by step
Series: 4, 6, 12, 30, 90, ? The differences are 2, 6, 18, 60, which is not clean. Try ratios: 6 ÷ 4 = 1.5; 12 ÷ 6 = 2; 30 ÷ 12 = 2.5; 90 ÷ 30 = 3. The multipliers increase by 0.5 each time: 1.5, 2, 2.5, 3, so the next is 3.5, giving 90 × 3.5 = 315. Test: 4 × 1.5 = 6 ✓; 6 × 2 = 12 ✓; 12 × 2.5 = 30 ✓; 30 × 3 = 90 ✓. The rule works for every term, so the answer is 315.
The lesson: when the differences look messy, look at the ratios, and when both are untidy, look at the differences of the differences or at alternate terms. Always test the rule on all the terms before you choose an option.
Common traps
- Fixing on the first pattern without testing on all terms.
- Missing alternating series when the differences look random.
- Arithmetic slips in squares and cubes.
- Forgetting the odd one out in "find the wrong number" questions.
Speed tips
- Memorise squares to 20 and cubes to 10. See squares, cubes and roots.
- Write the differences below the series in light pencil, and then the differences of the differences.
- Limit yourself to about 40 seconds; if no rule appears, mark and move on.
A practice routine
- Day 1: constant and changing differences, ten series.
- Day 2: multiplying and combined rules.
- Day 3: squares, cubes and alternating series.
- Day 4: "wrong number" series and a timed set of 12 at 40 seconds each.
Where it appears
Number series appear in banking reasoning and quantitative sections, SSC CGL reasoning, MAT Intelligence and Critical Reasoning and GATE General Aptitude. Related reasoning topics: coding-decoding and syllogisms.
Practice tests below are published by institutes and educators on TestMint.
