📚 Reasoning Ability – Series
Target Exam: Dadra & Nagar Haveli & Daman & Diu Police Constable
Topic: Series
1. Simple Explanation
A series is a list of numbers arranged in a particular order where a rule or pattern governs the progression from one term to the next.
In reasoning tests you are asked to:
- Identify the rule (arithmetic, geometric, Fibonacci, alternating, etc.).
- Find the missing term(s) or predict the next term(s).
The key is to look for regular changes (difference, ratio, addition/subtraction, multiplication/division, or a combination).
2. Types of Series & How to Spot Them
| Series Type | Pattern | How to Spot | Example |
|---|---|---|---|
| Arithmetic (A.P.) | Constant difference between consecutive terms | a, a+d, a+2d, ... |
2, 5, 8, 11 (d = +3) |
| Geometric (G.P.) | Constant ratio between consecutive terms | a, ar, ar², ... |
3, 6, 12, 24 (r = 2) |
| Fibonacci / Recursive | Each term = sum of two preceding terms | a, b, a+b, a+2b, ... |
1, 1, 2, 3, 5, 8 |
| Harmonic | Sequence of reciprocals of integers | 1, 1/2, 1/3, 1/4, ... |
1, 0.5, 0.333… |
| Alternating | Two (or more) different patterns interleaved | a, b, a+2, b-1, ... |
3, 7, 5, 11, 7, 15 |
| Square / Cube / Factorial | Terms are perfect squares/cubes/factorials | 1, 4, 9, 16 (n²) |
1, 8, 27, 64 (n³) |
| Prime / Composite | Sequence of primes or composites | 2, 3, 5, 7, 11 | 4, 6, 8, 9, 10 |
| Miscellaneous | Combination of operations, patterns in digits, etc. | 1, 12, 123, 1234, 12345 | 9, 18, 27, 36 |
3. Key Points to Remember
| # | Point | Why It Matters |
|---|---|---|
| 1 | Look for the simplest rule first (difference or ratio). | Easier and faster. |
| 2 | Check two consecutive differences or ratios to confirm a pattern. | Avoids mis‑identifying a pattern. |
| 3 | Consider alternating patterns when the series seems irregular. | Some series alternate +2, –1, +3, –2, etc. |
| 4 | Check for squares/cubes by taking square/cube roots. | Quick if numbers grow rapidly. |
| 5 | Look at the digits (e.g., 12, 123, 1234) – sometimes the rule is about concatenation. | Useful for “digit” series. |
| 6 | Use the position (n) of the term if the rule depends on n (e.g., n², 2n+1). | Helps in series like 3, 6, 12, 24 where nth term = 3×2ⁿ⁻¹. |
| 7 | Check for negative or zero terms – they may signal a different pattern. | Avoids sign errors. |
| 8 | Always verify the rule with all given terms before finalizing. | Prevents mistakes due to accidental matches. |
4. Worked Example
Problem:
Find the missing term in the series:
4, 9, 19, 39, ?, 147
Step‑by‑Step Solution
| Term | Value | Observation |
|---|---|---|
| 1 | 4 | – |
| 2 | 9 | 4 + 5 |
| 3 | 19 | 9 + 10 |
| 4 | 39 | 19 + 20 |
| 5 | ? | ? |
| 6 | 147 | ? |
-
Look at differences:
- 9 – 4 = 5
- 19 – 9 = 10
- 39 – 19 = 20
The differences are: 5, 10, 20 → Each difference is doubling (×2).
So the next difference should be 20 × 2 = 40. -
Compute missing term:
39 + 40 = 79. -
Verify with the last term:
79 + (next difference)
The next difference should be 40 × 2 = 80.
79 + 80 = 159, but the given last term is 147.
→ Our assumption of pure doubling is wrong. -
Try a different pattern:
Notice that 5, 10, 20 can be expressed as 5 × 1, 5 × 2, 5 × 4.
The multipliers are 1, 2, 4 → doubling each time.
Next multiplier should be 8 → difference = 5 × 8 = 40.
But as we saw, 39 + 40 = 79. Then 79 + 5 × 16 = 79 + 80 = 159, not 147. -
Alternate approach – maybe the pattern is: difference = previous term × 0.5
- 9 – 4 = 5 ≈ 4 × 1.25