Number Series
Subject: Mental Ability | Frequency: 2-4 questions per APPSC paper | Time: 45-60 sec/question
Introduction
Number series questions give a sequence of numbers following a hidden pattern. You must find the next term or identify the wrong term. The difference method alone solves 70%+ of questions.
Core Method
- Compute first-level differences (T2−T1, T3−T2, etc.)
- Constant difference? → Arithmetic series
- Constant ratio? (T2/T1 = T3/T2) → Geometric series
- Compute second-level differences if first aren't constant
- Check squares/cubes — n², n³, n²+k patterns
- Look for alternating patterns — split odd/even positions
- Check Fibonacci-type — each term = sum of two preceding
- Try mixed operations — ×2+1, ×3−2, etc.
Types with Examples
| Type | Pattern | Example | Answer |
|---|---|---|---|
| Arithmetic | Constant difference | 3, 7, 11, 15, ? (d=4) | 19 |
| Geometric | Constant ratio | 2, 6, 18, 54, ? (r=3) | 162 |
| Square | n² | 1, 4, 9, 16, 25, ? | 36 |
| Cube | n³ | 1, 8, 27, 64, ? | 125 |
| Fibonacci | Sum of previous two | 1, 1, 2, 3, 5, 8, ? | 13 |
| Prime | Consecutive primes | 2, 3, 5, 7, 11, 13, ? | 17 |
| Two-tier difference | Differences form a series | 1, 2, 5, 10, 17, 26, ? (diffs: 1,3,5,7,9→11) | 37 |
| Mixed operation | ×2+1 etc. | 2, 5, 11, 23, 47, ? | 95 |
| Wrong number | One doesn't fit | 2, 3, 5, 7, 11, 15, 17 → 15 is wrong | Should be 13 |
Worked Examples — Easy
Q1: 4, 8, 12, 16, 20, ? → d=4 → 24
Q2: 3, 9, 27, 81, ? → r=3 → 243
Q3: 1, 4, 9, 16, 25, 36, ? → 7² = 49
Worked Examples — Medium
Q4: 2, 6, 12, 20, 30, ? → Diffs: 4,6,8,10 → next=12 → 42 [Pattern: n(n+1)]
Q5: 3, 5, 9, 17, 33, ? → Diffs: 2,4,8,16 (geometric ×2) → next=32 → 65
Q6: 7, 11, 19, 35, ? → Diffs: 4,8,16 (doubling) → next=32 → 67
Worked Examples — Hard
Q7: 1, 5, 14, 30, 55, ? → Cumulative sums of squares → 55+36 = 91
Q8: 2, 12, 36, 80, 150, ? → n²(n+1): 1×2, 4×3, 9×4, 16×5, 25×6 → 36×7 = 252
Shortcuts & Tricks
| Shortcut | When to Use |
|---|---|
| Difference method | Try first on every series |
| Second difference | When first differences form a pattern |
| Ratio method | When terms grow rapidly |
| n² ± k | Terms close to perfect squares |
| Split sub-series | Alternating — separate odd/even positions |
| Reverse-engineer from options | In MCQs, test options backward |
Common Mistakes
- Not computing second-level differences
- Confusing arithmetic with geometric — check both difference AND ratio
- Missing alternating patterns — split odd/even positions
- Arithmetic errors in large numbers
- Assuming only one operation — mixed series use two or more operations
Exam Strategy
- Master the difference method first — it covers 70%+ of questions
- Mostly easy to medium; occasionally one hard question
- Time: 45-60 seconds per question
Practice Questions
- 5, 10, 20, 40, ? → 80 (×2)
- 2, 3, 5, 8, 12, ? → Diffs: 1,2,3,4→5 → 17
- 1, 8, 27, 64, 125, ? → 216 (6³)
- 4, 7, 12, 19, 28, ? → Diffs: 3,5,7,9→11 → 39
- 2, 5, 11, 23, 47, ? → ×2+1 → 95