Agniveer – Reasoning Ability
Topic: Series
Series questions test your ability to spot patterns and predict the next term(s).
They often appear in competitive exams (JEE, CAT, BITSAT, banking, etc.).
1. Simple Explanation
A series is a sequence of numbers (or symbols) where each element follows a rule.
The goal:
- Identify the rule (difference, ratio, position‑based, digit‑pattern, etc.).
- Use it to find the missing term or the next term(s).
2. Key Points (Quick Reference)
| # | What to Look For | How to Check | Example |
|---|---|---|---|
| 1 | Arithmetic Progression (AP) | Constant difference between consecutive terms. | 3, 7, 11, 15 → diff = +4 |
| 2 | Geometric Progression (GP) | Constant ratio between consecutive terms. | 2, 6, 18, 54 → ratio = ×3 |
| 3 | Fibonacci / Additive | Each term = sum of previous two (or other additive rule). | 1, 1, 2, 3, 5, 8 |
| 4 | Square / Cube / Factorial | Terms are perfect powers. | 1, 4, 9, 16 → squares |
| 5 | Alternating Pattern | Two (or more) interleaved subsequences. | 2, 5, 4, 11, 6, 17 (even positions → +3, odd → +2) |
| 6 | Digit / Positional Pattern | Rule depends on the digits, position, or both. | 12, 24, 36, 48 (sum of digits = 3, 6, 9, 12 → +3 each) |
| 7 | Combination of Operations | E.g., multiply then add, or add then square. | 1, 2, 6, 24 (×1, ×2, ×3, ×4) |
| 8 | Missing Number | Often a single term is omitted. Use patterns above to back‑track. | 2, 4, ___, 16 → AP → 8 |
| 9 | Multiple Rules | Sometimes two rules apply alternately. | 1, 2, 4, 3, 6, 5, 9… (×2, ×1.5, …) |
| 10 | Check for “Odd‑Even” or “Prime/Composite” | Some series use properties of numbers. | 2, 3, 5, 7, 11 → primes |
Tip: If you can’t spot a simple rule, try converting the series into differences or ratios first.
3. Worked Example
Question
Find the next term of the series:
5, 12, 20, 29, ___
Step‑by‑Step Solution
| Term | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 5 | 12 | 20 | 29 | ? |
| Difference | – | 7 | 8 | 9 | – |
Observation
- Differences are increasing by 1 each time: 7, 8, 9 → next difference = 10.
Calculation
Last term (29) + Next difference (10) = 39
Answer: 39
Why It Works
- The series is an AP of differences (a quadratic sequence).
- Recognizing the pattern in differences is often quicker than trying to fit a single rule to the terms.
4. Additional Practice Problems
| # | Series | Find |
|---|---|---|
| 1 | 2, 6, 12, 20, ___ | 30 |
| 2 | 3, 9, 27, ___ | 81 |
| 3 | 1, 4, 13, 40, ___ | 121 |
| 4 | 7, 14, 28, 56, ___ | 112 |
| 5 | 5, 7, 11, 13, 19, ___ | 23 |
(Work through them using the rules above!)
5. Quick Revision Summary
- Identify the pattern: Look for constant differences, ratios, or digit/position rules.
- Use differences/ratios: Convert the series into first differences or ratios; a simple pattern often emerges.
- Check for alternating subsequences: Split the series into odd/even positions and analyze separately.
- Handle missing numbers: Work backwards from known terms or use differences/ratios.
- Remember special sequences: AP, GP, Fibonacci, squares, cubes, primes.
- Practice with multiple rules: Some series mix operations; test each possibility.
- Verify: Once you find a rule, apply it to all terms to confirm consistency.
Keep practicing different series types; the more patterns you encounter, the faster you’ll spot them during exams!