Reasoning Ability – Series
For Assam Police Sub‑Inspector (SI) Exam
1. Simple Explanation
A series is a sequence of numbers arranged in a specific order. In reasoning tests you’ll be asked to:
- Identify the pattern (rule) that generates the series.
- Find the next term or any missing term.
- Determine the sum of a given number of terms.
- Check the validity of a proposed rule.
Typical series types:
| Type | Example | Key Idea |
|---|---|---|
| Arithmetic | 3, 7, 11, 15, … | Constant difference (d) |
| Geometric | 2, 6, 18, 54, … | Constant ratio (r) |
| Fibonacci‑like | 1, 1, 2, 3, 5, 8, … | Each term = sum of previous two |
| Square/Cube | 1, 4, 9, 16, … | (n^2) or (n^3) |
| Mixed/Pattern‑Based | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 | (n^2) but may be interleaved with other operations |
2. Key Points to Remember
| # | Point | Why It Matters |
|---|---|---|
| 1 | Check the first few terms | Most patterns can be deduced from 3–4 terms. |
| 2 | Look at successive differences | Helps spot arithmetic or second‑order patterns. |
| 3 | Check ratios | For geometric series. |
| 4 | Try squares/cubes | Many exam series use (n^2, n^3) or multiples of them. |
| 5 | Consider alternate terms | Some series use two interleaved patterns (odd/even). |
| 6 | Verify with a second method | Ensure the rule fits all given terms. |
| 7 | Remember “quick checks” | |
| 7a | Sum of first n terms of arithmetic: (S_n = n/2 (2a + (n-1)d)). | Saves time. |
| 7b | Sum of first n terms of geometric: (S_n = a(1-r^n)/(1-r)) if (r≠1). | Handy when the question asks for a sum. |
| 8 | Edge cases: 0, negative numbers, or missing terms in the middle. | Can change the pattern (e.g., alternating signs). |
| 9 | Time‑saving trick: If the series increases rapidly, check if it’s square/cube or exponential before trying a linear rule. | Avoids lengthy calculations. |
3. Worked Example
Question
Find the 7th term of the series:
14, 18, 23, 29, 36, …
Step 1: Look at Differences
- 18 – 14 = 4
- 23 – 18 = 5
- 29 – 23 = 6
- 36 – 29 = 7
Step 2: Identify the Pattern
Differences are increasing by 1 each time: 4, 5, 6, 7, …
So the k‑th difference (starting from k=1) is (k+3).
Step 3: Compute the Next Difference
For the 5th difference (between 6th and 7th terms):
(5 + 3 = 8)
Step 4: Find the 7th Term
- 7th term = 6th term + 8
- 6th term given as 36
- 7th term = 36 + 8 = 44
Answer: 7th term = 44
Quick Check
- 8th difference would be 9 → 44 + 9 = 53, plausible continuation.
- All differences follow the rule, so the series is consistent.
4. Quick Revision Summary
| Category | What to Do | Shortcut |
|---|---|---|
| Arithmetic | Find common difference (d). | (a_n = a_1 + (n-1)d) |
| Geometric | Find common ratio (r). | (a_n = a_1 r^{n-1}) |
| Second‑order (difference of differences) | Check if the first differences form an arithmetic series. | Use (\Delta^2 a_n = \text{constant}) |
| Squares/Cubes | Test if (a_n = n^2) or (n^3) (or multiples). | Quick mental check (1,4,9,16,25,… or 1,8,27,64,125…) |
| Alternating patterns | Split into two sub‑series (odd vs even positions). | Solve each sub‑series separately. |
| Missing term in the middle | Use the rule found to back‑calculate or forward‑calculate. | If you know 4th & 6th terms, you can find 5th by averaging (for arithmetic). |
| Sum questions | Plug into the sum formulas. | Use arithmetic/ geometric sum formulas. |
| Time‑saving | Always try the simplest pattern first (difference, ratio). | The simplest rule usually wins in exam time. |
Final Tips
- Always write down the first few terms to see patterns visually.
- Work backwards if the series is long; it’s sometimes easier to spot the rule from the tail.
- Double‑check that the rule fits all given terms, not just the first few.
- Practice diverse series types; the more you