Number Series
Overview
The task is simple: given a sequence of numbers following a hidden pattern, find the next term, a missing term, or identify the wrong term that breaks the pattern.
This topic tests your pattern recognition speed rather than heavy calculation. Success depends on quickly identifying whether the series follows addition, subtraction, multiplication, division, squares, cubes, primes, or a combination. Once you crack the underlying logic, solving becomes mechanical. Students who master the common series types can score full marks here with minimal time investment—making this a high-reward topic for exam preparation.
The key challenge is that TNPSC papers mix straightforward patterns with tricky variations. You must develop an instinct for spotting the difference type, ratio type, and alternating patterns within seconds.
Key Concepts
- Difference Series: Each term differs from the previous by a constant (arithmetic progression) or by a changing difference that itself forms a pattern.
- Ratio/Product Series: Each term is obtained by multiplying or dividing the previous term by a constant or varying factor.
- Square and Cube Series: Terms are perfect squares (1, 4, 9, 16...), perfect cubes (1, 8, 27, 64...), or derived by adding/subtracting from squares/cubes.
- Two-Tier Difference: When first differences are not constant, compute second differences—if those are constant, you have a quadratic pattern.
- Alternating Series: Two independent sub-series interleaved at odd and even positions. Separate them and solve individually.
- Mixed Operation Series: Pattern alternates between operations like +2, ×3, +2, ×3... Identify the cycle length.
- Prime Number Series: Terms are consecutive primes (2, 3, 5, 7, 11...) or differences between consecutive primes.
- Wrong Number Detection: One term violates the pattern. Find the rule, then identify which term does not fit.
Formulas / Key Facts
| Pattern Type | How to Identify | Example |
|---|---|---|
| Constant difference | Subtract consecutive terms; result is same | 5, 8, 11, 14 → difference = 3 |
| Increasing difference | Differences increase by fixed amount | 2, 3, 5, 8, 12 → diff: 1, 2, 3, 4 |
| Geometric (ratio) | Divide consecutive terms; ratio is constant | 3, 6, 12, 24 → ratio = 2 |
| Perfect squares | Terms = n² | 1, 4, 9, 16, 25 |
| Perfect cubes | Terms = n³ | 1, 8, 27, 64, 125 |
| n² + n pattern | Each term = position² + position | 2, 6, 12, 20 → 1²+1, 2²+2, 3²+3... |
| n² - 1 pattern | Terms = n² minus 1 | 0, 3, 8, 15, 24 |
| Prime series | Terms are prime numbers | 2, 3, 5, 7, 11, 13 |
| Fibonacci-type | Each term = sum of two preceding terms | 1, 1, 2, 3, 5, 8 |
Memorize these perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Memorize these perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
First 15 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Worked Examples
Example 1: Find the next term Series: 7, 11, 19, 35, 67, ?
Step 1: Find differences 11-7=4, 19-11=8, 35-19=16, 67-35=32 Differences: 4, 8, 16, 32
Step 2: Identify pattern in differences Each difference doubles (×2) Next difference = 32 × 2 = 64
Step 3: Add to last term 67 + 64 = 131
Example 2: Find the missing term Series: 2, 6, 12, 20, ?, 42
Step 1: Check if terms follow n² + n pattern Position 1: 1² + 1 = 2 ✓ Position 2: 2² + 2 = 6 ✓ Position 3: 3² + 3 = 12 ✓ Position 4: 4² + 4 = 20 ✓ Position 6: 6² + 6 = 42 ✓
Step 2: Apply pattern for position 5 5² + 5 = 25 + 5 = 30
Example 3: Find the wrong term Series: 3, 7, 15, 27, 63, 127
Step 1: Test pattern 2ⁿ - 1 2¹ - 1 = 1 (not 3, so adjust: maybe 2ⁿ + something)
Step 2: Try 2² - 1 = 3 ✓, 2³ - 1 = 7 ✓, 2⁴ - 1 = 15 ✓ 2⁵ - 1 = 31 (but series has 27 ✗) 2⁶ - 1 = 63 ✓, 2⁷ - 1 = 127 ✓
Answer: 27 is wrong; should be 31
Example 4: Alternating series Series: 3, 5, 9, 10, 27, 15, ?
Step 1: Separate odd and even positions Odd positions: 3, 9, 27 → multiplied by 3 (geometric) Even positions: 5, 10, 15 → adding 5 (arithmetic)
Step 2: Position 7 is odd Next in odd series: 27 × 3 = 81
Common Mistakes
- Assuming constant difference too quickly → Always compute at least 3-4 differences before concluding. Series may have second-level patterns.
- Ignoring alternating patterns → If differences seem random, separate odd/even positions and check each sub-series independently.
- Forgetting squares/cubes near boundaries → Students often miss that 49, 64, 81 or 125, 216 are nearby squares/cubes. Keep these memorized.
- Calculation errors in two-tier differences → Write differences neatly in a row below the series. Rushing causes sign errors or skipped terms.
- In wrong-term questions, assuming the last term is wrong → The wrong term can be anywhere. Verify the pattern using at least 4 terms before identifying the outlier.
Quick Reference
- First step: Always compute differences between consecutive terms.
- If differences are constant → Arithmetic Progression.
- If differences double/triple → Geometric-style growth in differences.
- Numbers like 2, 6, 12, 20, 30 → Think n(n+1) or n² + n.
- Random-looking series → Check for alternating sub-series at odd/even positions.
- Wrong term questions → Find the rule first, then test each term against it.