Number Series — Study Notes for Maharashtra Police Bharti (Constable)
Overview
Number Series is one of the most frequently asked reasoning topics in the Maharashtra Police Bharti exam.
This topic tests your ability to spot mathematical relationships quickly — addition, subtraction, multiplication, division, squares, cubes, or combinations thereof. Mastering number series boosts your speed in the entire reasoning section because the pattern-recognition skills transfer to other question types. The good news: with practice, most exam patterns fall into 8–10 recurring types, and you can crack them in under 30 seconds each.
Success here depends on two things — knowing the common pattern families and developing a systematic approach when the pattern isn't obvious at first glance.
Key Concepts
- Constant Difference Series: Each term increases or decreases by the same fixed number (e.g., +5, +5, +5).
- Increasing/Decreasing Difference Series: The difference itself grows or shrinks by a fixed amount (e.g., +2, +4, +6, +8).
- Multiplication/Division Series: Each term is obtained by multiplying or dividing the previous term by a constant (e.g., ×2, ×2, ×2).
- Square Series: Terms are perfect squares (1, 4, 9, 16, 25) or differences are squares.
- Cube Series: Terms are perfect cubes (1, 8, 27, 64, 125) or differences are cubes.
- Alternating Series: Two independent patterns interleaved at odd and even positions.
- Mixed Operation Series: Pattern uses different operations in sequence (e.g., +2, ×3, +2, ×3).
- Prime Number Series: Terms are consecutive prime numbers (2, 3, 5, 7, 11, 13).
Formulas / Key Facts
| Pattern Type | How to Identify | Example |
|---|---|---|
| Arithmetic (AP) | Constant difference between consecutive terms | 3, 7, 11, 15, 19 (diff = +4) |
| Geometric (GP) | Constant ratio between consecutive terms | 2, 6, 18, 54 (ratio = ×3) |
| Difference of differences | First differences form their own AP | 2, 3, 5, 8, 12 (diff = 1, 2, 3, 4) |
| Square-based | Terms near perfect squares | 1, 4, 9, 16, 25 |
| Cube-based | Terms near perfect cubes | 1, 8, 27, 64, 125 |
| Alternating | Separate odd-position and even-position sub-series | 2, 5, 4, 10, 6, 15 |
| n² ± n pattern | Each term = position² ± position | 2, 6, 12, 20 (n² + n) |
Must-know squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Must-know cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Worked Examples
Example 1 — Find the Next Term
Series: 5, 11, 23, 47, ?
Step 1: Calculate differences.
- 11 − 5 = 6
- 23 − 11 = 12
- 47 − 23 = 24
Step 2: Observe the differences — 6, 12, 24 (each is ×2 of previous).
Step 3: Next difference = 24 × 2 = 48.
Step 4: Next term = 47 + 48 = 95.
Example 2 — Find the Wrong Term
Series: 2, 5, 10, 17, (28), 37
One term is incorrect. Identify it.
Step 1: Check pattern — differences are 3, 5, 7, 11, 9.
Step 2: Expected differences should be consecutive odd numbers: 3, 5, 7, 9, 11.
Step 3: After 17, adding 9 gives 26, then adding 11 gives 37.
Step 4: The term 28 should be 26 — so 28 is the wrong term.
Example 3 — Find the Missing Term
Series: 3, 9, ?, 81, 243
Step 1: Check ratios — 9 ÷ 3 = 3, 243 ÷ 81 = 3.
Step 2: Pattern is ×3 (Geometric Progression).
Step 3: Missing term = 9 × 3 = 27.
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Assuming all series are simple AP | Always calculate at least 3–4 differences before deciding the pattern. |
| Ignoring second-level differences | When first differences aren't constant, compute differences of differences. |
| Not checking for alternating patterns | If pattern seems random, split into odd-position and even-position terms and check each separately. |
| Confusing squares and cubes | Keep a mental list of squares (up to 15²) and cubes (up to 10³) ready; verify before finalising. |
| Rushing without verification | After finding the pattern, apply it to all given terms to confirm before marking answer. |
Quick Reference
- First step always: Write down the differences between consecutive terms.
- Differences forming AP: Add the next logical difference to get the answer.
- Constant ratio between terms: Multiply by that ratio for the next term.
- When stuck: Check for squares, cubes, primes, or alternating sub-series.
- Wrong-term questions: The term that breaks the established pattern is the answer.
- Speed tip: Memorise squares 1–20 and cubes 1–10 — saves 10+ seconds per question.