Series in Reasoning Ability
For the Uttar Pradesh Patwari / Lekhpal Examination
Objective – Quickly identify the pattern of a numerical series and predict the next term, missing term, or sum of terms.
1. What is a “Series”?
A series is a sequence of numbers written in a specific order where each term follows a rule or pattern.
In reasoning tests the series may involve arithmetic progressions, geometric progressions, alternating patterns, prime numbers, Fibonacci‑type additions, and many combinations.
2. Common Types of Series
| Type | Definition | Typical Pattern | Example |
|---|---|---|---|
| Arithmetic (AP) | +k or –k each step | a, a+d, a+2d, … |
2, 5, 8, 11, … (d = +3) |
| Geometric (GP) | ×k or ÷k each step | a, ar, ar², … |
3, 6, 12, 24, … (r = 2) |
| Fibonacci | Sum of two preceding terms | a, b, a+b, a+2b, … |
1, 1, 2, 3, 5, 8 … |
| Prime Numbers | 2, 3, 5, 7, 11, … | No simple algebraic rule | |
| Square / Cube Numbers | n² or n³ | 1, 4, 9, 16, 25 … | 1, 8, 27, 64 … |
| Alternating Sign / Pattern | +, –, +, – … | 5, –3, 9, –7, … | |
| Multiple / Combination | Two or more sub‑series interleaved | 2, 4, 6, 8, 10, 12 … (two sub‑series of 2,4,6 & 4,8,12) | |
| Factorial / Exponential | n! or 2ⁿ | 1, 2, 6, 24, 120 … | 1, 4, 16, 64, … |
| Digit / Reversal | Reverse digits, sum of digits | 123, 321, 213, 312 … | |
| Pattern in Differences | Differences form a series | 3, 6, 12, 24, 48 … (diffs: 3,6,12,24…) |
Tip – If you can’t spot a simple rule, try looking at differences or ratios of adjacent terms. Those often reveal AP or GP hidden inside.
3. General Strategy to Solve a Series
| Step | What to Do | Why |
|---|---|---|
| 1. Write the series clearly | List every term, separate missing spots with a dash. | Clear view helps spot patterns. |
| 2. Check for a single‑step rule | Test AP (a₂-a₁), GP (a₂/a₁), or Fibonacci (a₃=a₁+a₂). |
Most series use one rule. |
| 3. Examine differences | Compute first‑order differences: Δa₁ = a₂-a₁, Δa₂ = a₃-a₂, … |
If differences form a series, you’re on the right track. |
| 4. Examine ratios | Compute a₂/a₁, a₃/a₂, … |
Useful for GP or exponential patterns. |
| 5. Look for alternation | Split the series into two (or more) interleaved sub‑series. | Alternating sequences often appear. |
| 6. Combine rules | Sometimes terms are a + b, a × b, or a² + b². |
For compound series, identify sub‑rules. |
| 7. Verify with multiple terms | Apply the suspected rule to a few terms to ensure consistency. | Avoids accidental coincidences. |
| 8. Predict | Use the confirmed rule to find the missing term, next term, or sum. | Final answer. |
4. Key Points to Remember
| Point | Explanation |
|---|---|
| Always start simple | Try AP or GP first before more complex patterns. |
| Use differences/ratios | They often reveal hidden AP/GP. |
| Check for alternation early | If the series oscillates (+ – + –), split it. |
| Watch for “double digits” | 12, 21, 23, 32, … may involve reversal or digit manipulation. |
| Prime & Fibonacci | Rarely used alone; often combined with another rule. |
| “What if it doesn’t fit? | Re‑examine the list; maybe a typo or a hidden relation. |
| Time Management | 1–2 minutes per series. If stuck, skip and return. |
| Practice | The more patterns you see, the faster you’ll spot them. |
5. Worked Example
Series – 2, 6, 12, 20, 30, 42, ?
Step‑by‑Step Solution
| Step | Action | Result |
|---|---|---|
| 1 | List | 2, 6, 12, 20, 30, 42, – |
| 2 | Check AP | 6–2 = 4, 12–6 = 6, … differences vary → not AP. |
| 3 | Check GP | 6/2 = 3, 12/6 = 2, … ratios vary → not GP. |
| 4 | Calculate first‑order differences | 4, 6, 8, 10, 12, … |
| 5 | Observe differences | They increase by 2 each |