Rajasthan Police Constable – Reasoning Ability
Topic: Series
Aim – To recognise, analyse and predict patterns in numerical or letter series.
1. Simple Explanation
A series is an ordered list of elements (numbers, letters, shapes, etc.) where a rule governs how one element follows the previous one.
In reasoning tests you’ll usually be asked to:
- Identify the rule (add, subtract, multiply, divide, alternate, etc.).
- Predict the next element or the element in an “x‑th” position.
- Fill the missing element(s).
2. Key Points (Quick Reference)
| Rule Type | Common Operations | Typical Pattern | Quick Check |
|---|---|---|---|
| Arithmetic | +K / –K | 2, 5, 8, 11 (K=3) |
Check difference between consecutive terms. |
| Geometric | ×R / ÷R | 3, 6, 12, 24 (R=2) |
Check ratio of consecutive terms. |
| Harmonic | 1/(a + K) | 1/2, 1/3, 1/4 |
Take reciprocals → arithmetic. |
| Fibonacci / Additive | a + b → next | 1, 1, 2, 3, 5, 8 |
Next = sum of previous two. |
| Alternating | Two separate rules for odd/even positions | 2, 3, 4, 5, 6, 7 → +1 each step |
Split series into two sub‑series. |
| Square / Cube / Nth Power | n², n³ | 1, 4, 9, 16 (n²) |
Identify by looking at differences: 3,5,7,… |
| Factorial / Double Factorial | n! | 1, 2, 6, 24 |
Recognise rapid growth. |
| Letter Series | Alphabetic steps, reverse, skip | A, C, E, G (+2) |
Convert to numbers (A=1, B=2, …). |
| Mixed Operations | +K, ×R alternately | 2, 6, 12, 30, 60 (+4, ×2) |
Look for repeating pattern of operations. |
| Mirror / Palindrome patterns | Reverse of earlier terms | 1, 2, 3, 2, 1 |
Check symmetry. |
Quick Diagnostic Checklist
- Check differences (Δ) → Arithmetic?
- Check ratios (Δ/prev) → Geometric?
- Check two‑term recurrence → Fibonacci?
- Check for two alternating sub‑series → Split.
- Look for squares/cubes → Compare Δ².
- For letters → Convert to numbers, then apply above steps.
3. Worked Example
Question
Series: 3, 9, 15, 27, 33, ?
Step‑by‑Step Solution
-
Identify sub‑series
Odd positions: 3, 15, 33
Even positions: 9, 27 -
Analyse odd sub‑series (1st, 3rd, 5th)
3 → 15 (+12)
15 → 33 (+18)
Differences: +12, +18 → increase by +6 each time.
Next difference expected: +24
So next odd term = 33 + 24 = 57. -
Analyse even sub‑series (2nd, 4th)
9 → 27 (×3).
Next even term would be 27 × 3 = 81, but we need the 6th term (even).
Since we only have two even terms, we assume the pattern continues:
27 × 3 = 81. -
Determine which position the blank is
Sequence:
1: 3 (odd)
2: 9 (even)
3: 15 (odd)
4: 27 (even)
5: 33 (odd)
6: ? (even) → 81
Answer: 81
Explanation – The series alternates between an arithmetic progression with increasing differences (+12, +18, …) for odd positions and a geometric progression (×3) for even positions. The 6th term is therefore 81.
4. Quick Revision Summary
- Arithmetic Series – constant difference (Δ).
- Geometric Series – constant ratio (R).
- Alternating Series – split into two sub‑series.
- Additive (Fibonacci‑type) – next = sum of previous two.
- Square/Cube Series – recognise by checking Δ² or rapid growth.
- Letter Series – map letters to numbers, then treat as numeric series.
- Mixed Operations – look for repeating patterns of +, –, ×, ÷.
- Check for Palindromic/ Mirror patterns – symmetry often indicates missing term.
Practice Tip
- Write down the first 3–4 terms.
- Try all the simple checks (Δ, R, sub‑series).
- If still stuck, test a few plausible operations (±, ×, ÷) on the first few terms and see if a pattern emerges.
Remember: In exam time, the quickest winning strategy is to spot the simplest rule first (usually Δ or R). Only move to more complex patterns if the first checks fail. Happy practising!