Tripura Forest Guard – Reasoning Ability
Topic: Series
Series are sequences of numbers (or letters, shapes, etc.) that follow a clear, repeatable rule.
In reasoning tests you’ll often be asked:
- Identify the rule
- Find the missing term
- Compute the n‑th term
- Calculate the sum of the first n terms
1. Simple Explanation
| Type | Formula / Rule | Example |
|---|---|---|
| Arithmetic Series | a, a+d, a+2d, … where d is a constant difference |
3, 7, 11, 15 … (a=3, d=4) |
| Geometric Series | a, ar, ar², … where r is a constant ratio |
2, 6, 18, 54 … (a=2, r=3) |
| Fibonacci / Recursive | aₙ = aₙ₋₁ + aₙ₋₂ |
1, 1, 2, 3, 5, 8 … |
| Alternating | Two interleaved patterns (e.g., +3, –2, +3, –2…) | 5, 8, 6, 9, 7, 10 … |
| Square / Cube Pattern | n², n³, … |
1, 4, 9, 16, … |
| Famous Pattern | 1, 1, 2, 3, 5, 8… (Fibonacci) | 0, 1, 1, 2, 3, 5, 8 … |
| Reverse or Mirror | Numbers reversed or in descending order | 9, 8, 7, 6, 5 … |
Key Idea:
Look for the smallest rule that can generate all given terms. If more than one rule fits, choose the one that yields a consistent pattern for the whole series.
2. Key Points to Remember
| # | What to Check | Why It Matters |
|---|---|---|
| 1 | Difference between consecutive terms | Detects arithmetic or alternating series |
| 2 | Ratio between consecutive terms | Detects geometric series |
| 3 | Sum of digits, factors, or parity | Detects digit‑based patterns (e.g., even/odd, prime, composite) |
| 4 | Repetition or cycle length | Recognizes patterns like “repeat every 3 terms” |
| 5 | Position index (n) | Helps in formulas like n² or n³ |
| 6 | Context clues (question wording) | Some series are “find the missing term” vs. “find the 10th term” |
| 7 | Check for “fancy” patterns (e.g., 1, 4, 9, 16, 25 = n²) |
Avoids missing simple solutions |
| 8 | Avoid over‑fitting | A rule that works for 3 terms but not 4 likely wrong |
3. Worked Example
Question:
Find the 6th term of the series:
2, 6, 12, 20, 30, …
Step‑by‑Step Solution
| Term | Value | Difference from previous term |
|---|---|---|
| a₁ | 2 | – |
| a₂ | 6 | +4 |
| a₃ | 12 | +6 |
| a₄ | 20 | +8 |
| a₅ | 30 | +10 |
| a₆ | ? | +12 (pattern continues) |
Observations
- Differences are 4, 6, 8, 10 – an arithmetic series with difference
2. - Hence the next difference should be
12.
Compute a₆
a₆ = a₅ + 12 = 30 + 12 = 42.
Answer: 42
Quick Verification
You can also spot that aₙ = n(n+1) for n = 1, 2, 3…
Plugging n = 6: 6*7 = 42. ✔️
4. Quick Revision Summary
| Topic | How to Approach | Common Mistakes |
|---|---|---|
| Identify the rule | Check differences → ratio → digit property → position | Looking only at first two terms |
| Arithmetic series | aₙ = a₁ + (n-1)d |
Forgetting that d could change (alternating) |
| Geometric series | aₙ = a₁ * r^(n-1) |
Mistaking ratio for difference |
| Alternating series | Break into 2 sub‑series | Assuming whole series is arithmetic |
| Recursive (Fibonacci) | aₙ = aₙ-1 + aₙ-2 |
Forgetting base cases |
| Square/Cube | Recognize n², n³ |
Missing the index shift (e.g., 1, 4, 9 is n² with n=1,2,3) |
| Sum of first n terms | Use formula: Sₙ = n/2 (2a₁ + (n-1)d) for arithmetic, Sₙ = a₁ (1-rⁿ)/(1-r) for geometric |
Not checking if r ≠ 1 |
| Missing term | Find rule, then back‑calculate | Applying wrong rule from earlier terms |
| Time‑saving tip | Write down differences/ratios first | Spending too long on one guess |
Final Tips for Exam Day
- Start with simple checks – differences and ratios first.
- Write it out – a quick table often reveals the pattern instantly.
- Use the first three terms – they usually indicate the rule.
- If stuck, try two likely rules