Mental Ability: Number Series — Patterns, Types, and Shortcuts

Number series is one of the most frequently tested mental ability topics in PSC exams. Expect 2-5 questions per paper. The key skill is pattern recognition — identifying the rule that connects consecutive terms.

Types of Number Series

Type 1: Arithmetic Series (Constant Difference)

The difference between consecutive terms is constant.

Pattern: a, a+d, a+2d, a+3d, …

Example	Series	Answer
1	3, 7, 11, 15, ?	19 (common difference = 4)
2	5, 13, 21, 29, ?	37 (common difference = 8)
3	100, 93, 86, 79, ?	72 (common difference = -7)

Type 2: Geometric Series (Constant Ratio)

Each term is multiplied by a constant to get the next.

Pattern: a, ar, ar², ar³, …

Example	Series	Answer
4	2, 6, 18, 54, ?	162 (ratio = 3)
5	5, 20, 80, 320, ?	1280 (ratio = 4)
6	1024, 512, 256, 128, ?	64 (ratio = 0.5, or dividing by 2)

Type 3: Difference Series (Increasing/Decreasing Differences)

Differences between terms form their own pattern.

Example	Series	Differences	Answer
7	2, 3, 5, 8, 12, ?	1, 2, 3, 4, 5	17
8	1, 2, 4, 7, 11, 16, ?	1, 2, 3, 4, 5, 6	22
9	3, 4, 7, 12, 19, 28, ?	1, 3, 5, 7, 9, 11	39

Strategy: When simple differences don’t show a pattern, take second-level differences (differences of differences).

Type 4: Square and Cube Series

Terms are perfect squares or cubes, or related to them.

Example	Series	Pattern	Answer
10	1, 4, 9, 16, 25, ?	1², 2², 3², 4², 5², 6²	36
11	1, 8, 27, 64, 125, ?	1³, 2³, 3³, 4³, 5³, 6³	216
12	2, 5, 10, 17, 26, ?	1²+1, 2²+1, 3²+1, 4²+1, 5²+1, 6²+1	37
13	0, 3, 8, 15, 24, ?	1²-1, 2²-1, 3²-1, 4²-1, 5²-1, 6²-1	35
14	2, 9, 28, 65, 126, ?	1³+1, 2³+1, 3³+1, 4³+1, 5³+1, 6³+1	217

Type 5: Multiplication/Operation Series

Each term is derived by performing an operation on the previous term.

Example	Series	Pattern	Answer
15	2, 4, 12, 48, ?	×2, ×3, ×4, ×5	240
16	1, 1, 2, 6, 24, ?	×1, ×2, ×3, ×4, ×5	120 (Factorials!)
17	3, 6, 18, 72, ?	×2, ×3, ×4, ×5	360

Type 6: Fibonacci-Type Series

Each term is the sum of the two preceding terms.

Example	Series	Pattern	Answer
18	1, 1, 2, 3, 5, 8, ?	Fibonacci: each = sum of previous two	13
19	2, 3, 5, 8, 13, 21, ?	Same pattern, different start	34
20	1, 4, 5, 9, 14, 23, ?	Each = sum of previous two	37

Type 7: Alternating Series

Two separate patterns alternate in one series.

Example	Series	Pattern	Answer
21	1, 2, 3, 6, 5, 18, 7, ?	Odd positions: 1,3,5,7 (+2); Even positions: 2,6,18,54 (×3)	54
22	3, 5, 9, 11, 15, 17, ?	+2, +4, +2, +4, +2, +4	21
23	2, 8, 3, 27, 4, 64, 5, ?	Odd: 2,3,4,5; Even: 8=2³, 27=3³, 64=4³, 5³	125

Type 8: Prime Number Series

Terms are prime numbers or derived from primes.

Example	Series	Pattern	Answer
24	2, 3, 5, 7, 11, 13, ?	Consecutive primes	17
25	4, 9, 25, 49, 121, ?	2², 3², 5², 7², 11², 13²	169
26	3, 5, 11, 13, 17, 23, ?	Prime pairs (twin primes pattern)	29

Remember first 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71

Type 9: Mixed Operation Series

Different operations applied in sequence.

Example	Series	Pattern	Answer
27	2, 3, 6, 7, 14, 15, ?	+1, ×2, +1, ×2, +1, ×2	30
28	3, 4, 12, 13, 39, 40, ?	+1, ×3, +1, ×3, +1, ×3	120

Type 10: Wrong Number in Series

Find the number that doesn’t fit the pattern.

Example	Series	Wrong Number	Correct
29	2, 5, 10, 17, 24, 37	24 (should be 26: n²+1 series)	26
30	1, 4, 9, 15, 25, 36	15 (should be 16: perfect squares)	16

Shortcut Strategies for PSC Exams

Step-by-Step Approach

1. Check differences — subtract consecutive terms. If constant, it is arithmetic.
2. Check ratios — divide consecutive terms. If constant, it is geometric.
3. Check differences of differences — if first differences form a pattern, it is Type 3.
4. Check for squares/cubes — see if terms relate to n², n³, n²+1, n²-1, etc.
5. Check alternating positions — separate odd and even positioned terms.
6. Check for primes — are the numbers prime or related to primes?
7. Check operations — ×2+1, ×3-1, or similar patterns.

Key Number Patterns to Memorize

n	n²	n³
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1000
11	121	1331
12	144	1728
13	169	2197
14	196	—
15	225	—
16	256	—
17	289	—
18	324	—
19	361	—
20	400	—
25	625	—

Factorial Values

n	n!
1	1
2	2
3	6
4	24
5	120
6	720
7	5040

Practice Problems

Solve these yourself, then check the answers below.

1. 4, 9, 16, 25, ?
2. 3, 6, 12, 24, 48, ?
3. 1, 4, 9, 16, 25, 36, ?
4. 2, 6, 12, 20, 30, ?
5. 7, 11, 13, 17, 19, 23, ?
6. 1, 3, 7, 15, 31, ?
7. 5, 10, 13, 26, 29, 58, ?
8. 120, 60, 20, 5, ?
9. 2, 12, 36, 80, 150, ?
10. 3, 5, 9, 17, 33, ?

Answers

1. 36 (perfect squares: 2², 3², 4², 5², 6²)
2. 96 (geometric, ratio = 2)
3. 49 (perfect squares: 1² to 7²)
4. 42 (differences: 4, 6, 8, 10, 12 — or n(n+1): 1×2, 2×3, 3×4, 4×5, 5×6, 6×7)
5. 29 (consecutive primes starting from 7)
6. 63 (×2+1: 1×2+1=3, 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63)
7. 61 (alternating: ×2, +3, ×2, +3, ×2, +3)
8. 1 (÷2, ÷3, ÷4, ÷5)
9. 252 (n²(n+1): 1²×2=2, 2²×3=12, 3²×4=36, 4²×5=80, 5²×6=150, 6²×7=252)
10. 65 (each term = previous × 2 - 1: 3×2-1=5, 5×2-1=9, 9×2-1=17, 17×2-1=33, 33×2-1=65)

Exam Tips

• Time management: Spend no more than 45-60 seconds per series question.
• Start with differences: This solves 60% of series questions instantly.
• Know your squares and cubes up to 20: This is non-negotiable for speed.
• If stuck, try separating odd/even positions: Alternating series are common tricks.
• Wrong number questions: Apply the pattern to each term and find the one that breaks it.