📚 Study Notes – Series (Reasoning Ability)
Target Exam: Goa Police Constable – Reasoning Ability
Module: Series
1️⃣ Simple Explanation
A series is a set of numbers arranged in a particular order where a rule (or pattern) governs the transition from one term to the next. In reasoning tests you’ll encounter:
| Pattern Type | Example | Rule |
|---|---|---|
| Arithmetic | 3, 7, 11, 15 | Add the same constant ( +4 ) |
| Geometric | 2, 6, 18, 54 | Multiply by the same constant ( ×3 ) |
| Fibonacci‑like | 1, 1, 2, 3, 5, 8 | Each term = sum of two previous terms |
| Alternating | 4, 9, 4, 9 | Two sub‑series interleaved |
| Mixed | 2, 5, 10, 17, 26 | Add 3, add 5, add 7, add 9, … |
| Consecutive | 1, 4, 10, 20 | n² + n |
Your job is to spot the rule and predict the missing term(s).
2️⃣ Key Points & Tips
| # | Key Point | Why It Matters |
|---|---|---|
| 1 | Identify the operation (addition, subtraction, multiplication, division, or a combination). | Determines the pattern family. |
| 2 | Check for alternating patterns (even/odd positions). | Many series use two interleaved sub‑series. |
| 3 | Look for simple arithmetic or geometric progressions first. | They are the most common and easiest to spot. |
| 4 | Test the first few terms with possible operations (e.g., +1, ×2, +2², etc.). | Quick validation of the rule. |
| 5 | Avoid over‑complicating – if two simple patterns fit, choose the simpler one. | Saves time and reduces errors. |
| 6 | Remember special patterns: Fibonacci, squares, cubes, primes, etc. | Recognise quickly in tests. |
| 7 | Check if the series is “increasing” or “decreasing”. | Negative or fractional patterns are rare. |
| 8 | Use a small table to write down the terms and compute differences or ratios. | Visual helps you see patterns. |
| 9 | Practice with mixed series – learn to spot the “next difference” trend. | Common in higher‑level tests. |
| 10 | Scrutinise the answer choices – sometimes only one choice fits all rules. | Eliminates guesswork. |
3️⃣ Worked Example
Question
Series: 3, 8, 15, 26, 41, ?
Step‑by‑Step Solution
| Term | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Value | 3 | 8 | 15 | 26 | 41 | ? |
-
Compute successive differences
8−3 = 5
15−8 = 7
26−15 = 11
41−26 = 15Differences: 5, 7, 11, 15
-
Look for a pattern in the differences
Differences themselves seem to increase by 2, 4, 4.
Alternatively, notice they are prime numbers: 5, 7, 11, 15 (but 15 isn’t prime).
So that route is inconsistent. -
Try a pattern on the differences of differences
7−5 = 2
11−7 = 4
15−11 = 4
Differences of differences: 2, 4, 4 → after the first jump, they stay +4. -
Predict the next difference
After 15, add 4 → next difference = 19. -
Add this to the last term
41 + 19 = 60.
Answer: 60
Verification
Check the whole series with the derived rule:
- Start with 3.
- Add 5 → 8.
- Add 7 → 15.
- Add 11 → 26.
- Add 15 → 41.
- Add 19 → 60.
The increments follow the pattern +5, +7, +11, +15, +19 (each time adding 2, 4, 4, 4).
4️⃣ Quick Revision Summary
| Pattern | Typical First Few Terms | Common Rule | Quick Check |
|---|---|---|---|
| Arithmetic | 4, 9, 14, 19 | +5 each | Difference constant |
| Geometric | 2, 6, 18, 54 | ×3 each | Ratio constant |
| Fibonacci‑like | 1, 1, 2, 3, 5, 8 | Sum of two prev | next = sum(prev₂) |
| Alternating | 5, 10, 5, 10 | Two sub‑series | check even/odd |
| Mixed addition | 1, 4, 9, 16 | Add successive odd numbers | Δ = 3,5,7,… |
| Mixed multiplication | 3, 6, 12, 24 | ×2 each | Ratio 2 |
| Square‑based | 2, 6, 12, 20 | n² + n | formula n²+n |
| Prime‑based | 2, 3, 5, 7 |