Study Notes – Reasoning Ability: Series
(For Goa Patwari / Lekhpal Examination – Program)
1. What is a “Series”?
A series is a sequence of numbers arranged in a particular order.
In reasoning tests, you are usually asked to identify the rule that governs the
sequence and then:
- Predict the next term
- Find a missing term
- Determine the sum of a given number of terms
- Spot a pattern in the differences / ratios
Tip: “Series” problems are all about recognizing the hidden rule.
2. Common Types of Series & Their Characteristics
| Type | Rule (formula) | Example | Key Feature |
|---|---|---|---|
| Arithmetic (A.P.) | (a_n = a_1 + (n-1)d) | 3, 7, 11, 15, … | Constant difference (d) |
| Geometric (G.P.) | (a_n = a_1 \times r^{,n-1}) | 2, 6, 18, 54, … | Constant ratio (r) |
| Fibonacci / Recursive | (a_n = a_{n-1} + a_{n-2}) | 1, 1, 2, 3, 5, 8, … | Each term = sum of two previous |
| Alternating Series | Alternate between two rules (e.g., +2 / –3) | 5, 7, 4, 6, 3, 5, … | Two interleaved patterns |
| Square / Cube / Factorial | (n^2, n^3, n!) | 1, 4, 9, 16, … | Powers of numbers |
| Prime / Composite | Sequence of primes or composites | 2, 3, 5, 7, 11, … | Prime numbers |
| Factorial / Double Factorial | (n!) or (n!!) | 1, 2, 6, 24, 120, … | Products of consecutive integers |
| Mixed / “Special” | Combination of arithmetic/geometry + add/subtract | 3, 6, 11, 18, 27, … | No single simple rule |
Quick Check:
- Look at the differences (Δ). Constant Δ → A.P.
- Look at the ratios (×). Constant ratio → G.P.
- Look for “add previous two” → Fibonacci.
- Look for repeated “plus/minus” patterns → Alternating.
3. How to Tackle a Series Question?
| Step | What to Do | Why It Helps |
|---|---|---|
| 1. Write the sequence | List the given numbers clearly. | Avoids missing a term. |
| 2. Check differences (Δ) | Compute successive differences. | Constant Δ = A.P.; Δ itself forms a pattern → higher‑order series. |
| 3. Check ratios (×) | Divide successive terms. | Constant ratio = G.P.; ratio forms a pattern → higher‑order series. |
| 4. Look for add/subtract patterns | Check if terms alternate in sign or magnitude. | Detects alternating series. |
| 5. Try small (n) values | Plug into potential formulas. | Confirms or rejects guessed rule. |
| 6. Validate with given terms | Re‑generate the sequence using your rule. | Ensures no mistake. |
| 7. Compute the answer | Use formula or pattern to find missing/next term. | Final answer. |
4. Worked Example
Question (Typical):
Find the 8th term of the series:
2, 6, 12, 20, 30, …
Solution
| n | Term | Δ (difference) |
|---|---|---|
| 1 | 2 | – |
| 2 | 6 | 4 |
| 3 | 12 | 6 |
| 4 | 20 | 8 |
| 5 | 30 | 10 |
- Observation: Differences are 4, 6, 8, 10 → increasing by 2 each time.
- Interpretation: Δ itself is an A.P. with first term 4 and common difference 2.
- General Δ formula: (Δ_n = 4 + 2(n-1) = 2n + 2).
- Find Δ for n = 5 → 2·5 + 2 = 12.
- Next term (n=6): (a_6 = a_5 + Δ_5 = 30 + 12 = 42).
- Continue to n=7: Δ_6 = 14 → (a_7 = 42 + 14 = 56).
- n=8: Δ_7 = 16 → (a_8 = 56 + 16 = 72).
Answer: 72
Verification by formula
We can also deduce a direct formula.
The sequence is (a_n = n^2 + n).
For n=8: (8^2 + 8 = 64 + 8 = 72). ✔️
5. Quick Revision Summary
| Category | Key Idea | Quick Test |
|---|---|---|
| Arithmetic | Constant Δ | If Δ constant → A.P. |
| Geometric | Constant × | If ratio constant → G.P. |
| Fibonacci / Recursive | Each term = sum of two previous | Check last two terms |
| Alternating | Two interleaved patterns | Split list into odd/even positions |
| Power Series | (n^2, n^3, …) | Check if each term is a perfect square/cube |
| Prime/Composite | Sequence of primes/composites | List primes: 2,3,5, |