📚 Study Notes – Reasoning Ability
Topic: Series
“Series” is a sequence of numbers (or letters, shapes, etc.) that follows a definite rule.
In the J&K Police Constable exam you’ll be asked to identify the rule and continue the sequence or find a missing term.
1️⃣ Simple Explanation
- Definition – A series is a list of elements where each element is obtained by applying a fixed rule to its predecessor(s).
- Why It Matters – Exams test your ability to spot patterns quickly.
- Common Formats –
- Arithmetic (constant difference)
- Geometric (constant ratio)
- Fibonacci‑type (sum of previous terms)
- Square/Cube, Prime, Factorial, etc.
- Mixed or “hidden” rules (digit manipulation, alternating patterns, etc.)
2️⃣ Key Points & Quick Checklist
| Step | What to Look For | How to Check |
|---|---|---|
| 1. Write the series clearly | List each term in a row or column | E.g., 2, 4, 8, 16 |
| 2. Compute first differences | Subtract consecutive terms | 4‑2 = 2, 8‑4 = 4 |
| 3. Compute second differences | Differences of differences | 4‑2 = 2 |
| 4. Check ratios | Divide consecutive terms | 4/2 = 2 |
| 5. Look for alternating patterns | Odd‑even positions | 1, 3, 5 vs 2, 4, 6 |
| 6. Test special sequences | Squares, cubes, primes, factorials | 1², 2², 3²… |
| 7. Check digit operations | Sum of digits, product, reverse | 12 → 1+2=3 |
| 8. Confirm with several terms | Rule should hold for 3+ terms | If not, try another rule |
Remember:
- Always start with simple patterns (differences/ratios).
- If both differences and ratios are constant → Arithmetic and Geometric simultaneously? (rare, e.g., 0,0,0… or 1,1,1…).
3️⃣ Types of Series & Typical Formulas
| Series Type | Rule (Formula) | Example | Next Term |
|---|---|---|---|
| Arithmetic (A.P.) | (a_n = a_1 + (n-1)d) | 3, 7, 11, 15 | 19 |
| Geometric (G.P.) | (a_n = a_1 \times r^{,n-1}) | 2, 4, 8, 16 | 32 |
| Fibonacci‑type | (a_n = a_{n-1} + a_{n-2}) | 1, 1, 2, 3, 5 | 8 |
| Squares | (a_n = n^2) | 1, 4, 9, 16 | 25 |
| Cubes | (a_n = n^3) | 1, 8, 27, 64 | 125 |
| Factorial | (a_n = n!) | 1, 2, 6, 24 | 120 |
| Prime Numbers | (a_n =) nth prime | 2, 3, 5, 7, 11 | 13 |
| Triangular Numbers | (a_n = n(n+1)/2) | 1, 3, 6, 10 | 15 |
| Alternating Add/Subtract | (a_{n} = a_{n-1} \pm k) | 5, 8, 5, 8 | 5 |
| Digit Sum Pattern | Sum of digits = constant or increases | 12, 21, 30 (1+2=3, 2+1=3, 3+0=3) | 39 (3+9=12) |
4️⃣ Worked Example
Question:
Find the next term of the series:
5, 12, 23, 38, 57, ?
Step‑by‑Step
-
Write the terms clearly
5 | 12 | 23 | 38 | 57 -
Compute first differences