📚 Reasoning Ability – Series
(For Punjab Patwari / Lekhpal Examination)
1. Simple Explanation
In reasoning tests, a series is a sequence of numbers arranged in a specific order.
The test‑taker must identify the rule that generates the series and then determine:
- Next term(s)
- Missing term(s)
- Sum of a certain number of terms
Knowing the main types of series and how to spot their patterns is essential.
2. Key Points
| Type | Common Pattern | Formula (if any) | Example |
|---|---|---|---|
| Arithmetic Series (AP) | Each term = previous term + constant difference d | a_n = a_1 + (n-1)d |
2, 5, 8, 11, … (d = 3) |
| Geometric Series (GP) | Each term = previous term × constant ratio r | a_n = a_1 * r^(n-1) |
3, 6, 12, 24, … (r = 2) |
| Fibonacci (or similar) | Each term = sum of previous two terms | a_n = a_(n-1) + a_(n-2) |
1, 1, 2, 3, 5, 8, … |
| Alternating Series | Pattern changes every other term (±, ×, ÷, etc.) | — | 1, 3, 6, 8, 13, 15, … |
| Square / Cube Pattern | Terms involve perfect squares/cubes or their roots | a_n = n^2 or a_n = n^3 |
1, 4, 9, 16, 25, … |
| Prime / Composite Pattern | Sequence of primes, composites, or differences thereof | — | 2, 3, 5, 7, 11, 13, … |
| Digit/Position Patterns | Numbers formed by manipulating digits (e.g., 12 ➜ 21) | — | 12, 21, 13, 31, … |
| Sum of Digits or Factors | Term equals sum of digits, factors, etc. | — | 12 (1+2=3), 15 (1+5=6) … |
Common Reasoning Tactics
| Tactic | What to look for | Example |
|---|---|---|
| Check differences | If differences are constant → AP; if differences form a pattern → other series | 2, 5, 10, 17 (diffs: 3,5,7) |
| Check ratios | If ratios are constant → GP; if ratios alternate → alternating series | 4, 8, 12, 24 (ratios: 2,1.5,2) |
| Check sums of digits | If digits add up to a consistent number | 13 (1+3=4), 22 (2+2=4) |
| Check positions (odd/even) | Separate odd and even positions may follow different rules | 1, 4, 9, 16, 25 (odd positions 1,9,25 …) |
| Look for “reverse” pattern | The series might mirror itself | 2, 4, 6, 8, 6, 4, 2 |
| Use “±” or “×” alternation | Alternating operations → separate two sub‑series | 3, 6, 9, 12, 15, 18 (add 3 each time) |
| Check for “missing terms” | Verify if a term is omitted to test pattern recognition | 2, 5, ?, 11 (missing 8) |
3. Worked Example
Question:
Find the missing number in the series:
4, 7, 12, 19, ?, 37, 52
Solution Steps
- Compute consecutive differences
| Term | Difference to next |
|---|---|
| 4 → 7 | +3 |
| 7 → 12 | +5 |
| 12 → 19 | +7 |
| 19 → ? | +9 (pattern: +3, +5, +7, +9…) |
| ? → 37 | +? |
| 37 → 52 | +15 |
-
Apply the pattern
- The differences increase by 2 each time: 3, 5, 7, 9, 11, 13, 15.
- So, after 19 the difference should be +9 →
19 + 9 = 28.
-
Verify
- Next difference should be +11:
28 + 11 = 39. - But the series says next term is 37.
- Our assumption is wrong → maybe the pattern alternates.
- Next difference should be +11:
-
Try an alternating pattern
-
Separate odd and even positions:
Odd positions: 4, 12, ?, 52
Even positions: 7, 19, 37 -
Odd: difference 8, (?), 40 → not clear.
-
Even: difference 12, 18 → increments by 6.
-
-
Re‑evaluate
-
Perhaps the series is sum of two series:
4+7=11,12+19=31,?+37=??,52+?… This is too complex.
-
-
Check for a simple rule:
-
Notice the series may be “add consecutive odd numbers”:
4 + 3 = 7
7 + 5 = 12
12 + 7 = 19
19 + 9 = 28
28 + 11 = 39 (but next is 37) → wrong.
-
-
Alternative: Subtract consecutive even numbers
- 7 - 4 = 3
- 12 - 7 = 5
- 19 - 12 = 7
- ? - 19 = 9 → ? = 28
- 37 - ? = 9? → 37-