Sikkim PSC – Reasoning Ability
Topic: Series
Series – a set of numbers arranged in a particular order, where each number follows a definite rule or pattern.
In reasoning tests you’ll often be asked to find the missing term, the next term, or to identify the rule governing a given series.
1. Simple Explanation
| Type | Definition | Common Pattern | Example |
|---|---|---|---|
| Arithmetic Series (AP) | Constant difference between consecutive terms | a, a+d, a+2d, … |
3, 7, 11, 15, … (difference = 4) |
| Geometric Series (GP) | Constant ratio between consecutive terms | a, ar, ar², … |
2, 6, 18, 54, … (ratio = 3) |
| Fibonacci / Additive Series | Each term = sum of one or more previous terms | Fₙ = Fₙ₋₁ + Fₙ₋₂ |
1, 1, 2, 3, 5, 8, … |
| Harmonic Series | Reciprocal of natural numbers | 1, 1/2, 1/3, 1/4, … |
2, 1, 2/3, 1/2, … |
| Alternating Series | Terms alternate between two or more patterns | a, b, a, b, … |
5, 10, 5, 10, … |
| Difference‑of‑Squares | Each term is expressed as difference of two squares | n² – (n‑1)² |
3, 7, 13, 21, … (3=2²‑1², 7=3²‑2², …) |
| Factorial / Multiplication Series | Each term = previous term × constant factor | 4, 20, 120, … (×5, ×6, …) |
|
| Digit‑Sum / Reverse‑Digit | Pattern involves digits of the number | 12, 21, 12, 21, … |
123, 321, 123, … |
Other specialised series:
Triangular numbers, square numbers, cubes, prime numbers, powers of 2/3, etc.
2. Key Points to Spot the Pattern
| Step | What to Look For |
|---|---|
| 1. Check the first few terms | Are they increasing or decreasing? |
| 2. Compute differences | Constant difference → arithmetic; varying → other patterns. |
| 3. Compute ratios | Constant ratio → geometric; else, look for other multiplicative patterns. |
| 4. Look for digit manipulation | Reversals, sum of digits, product of digits, alternating digits. |
| 5. Check for additive patterns | Each term = sum of previous one or two terms. |
| 6. Identify special number sequences | 1, 3, 6, 10… (triangular), 1, 4, 9, 16… (perfect squares). |
| 7. Combine patterns | Sometimes two rules are combined (e.g., AP inside GP). |
| 8. Cross‑validate | Verify the identified rule with all given terms. |
Tip: Write the first 4–5 terms in a table and compute differences/ratios side‑by‑side; patterns usually become obvious.
3. Worked Example
Question:
Find the 5th term of the series: 2, 6, 14, 30, …
Step‑by‑Step
| Term | Value | Difference from previous term |
|---|---|---|
| a₁ | 2 | – |
| a₂ | 6 | 6 – 2 = 4 |
| a₃ | 14 | 14 – 6 = 8 |
| a₄ | 30 | 30 – 14 = 16 |
Observation:
Differences are 4, 8, 16 – each difference is twice the previous one (ratio 2).
So the difference sequence itself is a geometric series with first term 4 and ratio 2.
Hence the next difference:
16 × 2 = 32
Now add this to the last known term:
30 + 32 = 62
Answer: The 5th term is 62.
Why This Works
- The series is not a simple AP or GP.
- Its differences form an AP with a common ratio (geometric).
- By finding the pattern in the difference sequence, we can extend the original series.
4. Quick Revision Summary
| Series Type | Rule | Quick Check | Example |
|---|---|---|---|
| Arithmetic (AP) | aₙ = a₁ + (n‑1)d |
Constant difference | 3, 8, 13, … (d=5) |
| Geometric (GP) | aₙ = a₁·rⁿ⁻¹ |
Constant ratio | 2, 4, 8, … (r=2) |
| Fibonacci / Additive | aₙ = aₙ₋₁ + aₙ₋₂ |
Sum of 2 previous | 1, 1, 2, 3, 5, … |
| Digit‑Based | Manipulate digits | Reversal, sum of digits | 12, 21, 12, … |
| Alternating | Two interleaved patterns | Odd vs even positions | 5, 10, 5, 10, … |
| Difference‑of‑Squares | n² – (n‑1)² |
2n‑1 |
3, 7, 13, 21, … |
| Special Number Series | Triangular, squares, primes, etc. | Recognise known sequences | 1, 4, 9, 16, … |
Remember:
- Write it out.
- Compute differences/ratios.
- Look for digit tricks.