Number Series Patterns: Civil Service Exam Guide

Number series questions are among the most common and scoring-friendly items on the Civil Service Exam. They test your ability to recognize patterns, apply numerical logic, and predict the next value based on mathematical relationships. While many examinees find this section intimidating, most number series follow predictable rules—once you understand these patterns, the questions become much easier to solve.

This guide provides a complete overview of the most frequently used number series patterns, step-by-step solving strategies, and sample problems that reflect the Civil Service Exam style. By the end of this guide, you will have the confidence to approach any sequence with clarity and accuracy.

What Are Number Series?

A number series is a set of numbers arranged in a specific order that follows a hidden rule or pattern. Your task is to identify the rule and determine the next number (or missing number) in the sequence.

Common number series formats include:

• Increasing or decreasing sequences

• Alternating patterns

• Patterns involving arithmetic or geometric operations

• Mixed operations

• Square, cube, or power-based progressions

• Fibonacci or additive patterns

• Difference-based patterns

Understanding these categories allows you to quickly analyze and classify the series in front of you.

Why Number Series Appears in the Civil Service Exam

The Civil Service Exam includes number series questions to evaluate:

• Numerical reasoning

• Pattern recognition

• Logical thinking

• Attention to detail

• Ability to analyze data quickly

These skills are essential for government work that involves interpreting information, solving problems, and making decisions under time pressure.

Number series questions are typically 10–15 items, depending on the exam type, and can significantly boost your score if you know the common patterns.

General Strategy for Solving Number Series

Before learning specific types of patterns, you must master a universal approach that applies to all number sequences.

1. Look for Differences Between Numbers

Subtract each term from the next. If the differences form a pattern (e.g., +2, +4, +6), you’re likely dealing with an arithmetic or incremental series.

2. Check for Multiplication or Division

If the numbers grow or shrink quickly, check for multiplication patterns (×2, ×3, ×1.5, etc.).

3. Examine Combined Operations

Some series use a combination (e.g., ×2 + 1).

4. Identify Alternating Patterns

Not all sequences are linear. Some alternate between two or more rules.

5. Consider Square or Cube Numbers

Look for sequences involving powers, like 2², 3², 4², or 2³, 3³, 4³.

6. Examine Fibonacci-Type Patterns

A common pattern involves adding the previous two numbers.

7. Always Verify Your Hypothesis

Check whether your assumed pattern works consistently across the entire sequence.

Common Number Series Patterns in the Civil Service Exam

Below are the patterns you must master, along with explanations and examples.

Arithmetic Series

An arithmetic series changes by adding or subtracting a constant number.

Example:
2, 6, 10, 14, 18, ___
Pattern: +4
Answer: 22

Another example:
50, 45, 40, 35, ___
Pattern: –5
Answer: 30

Arithmetic series are the simplest to solve and appear frequently in the exam.

Geometric Series

A geometric series multiplies or divides by a constant ratio.

Example:
3, 9, 27, 81, ___
Pattern: ×3
Answer: 243

Another example:
64, 32, 16, 8, ___
Pattern: ÷2
Answer: 4

These sequences grow or shrink rapidly, making them easy to spot.

Incremental or Variable-Difference Series

Some sequences use a changing difference (e.g., +1, +2, +3 …).

Example:
1, 3, 6, 10, 15, ___
Differences: +2, +3, +4, +5
Answer: Next difference is +6 → 21

Another variation:
4, 7, 12, 19, 28, ___
Differences: +3, +5, +7, +9
Next difference: +11 → 39

These are extremely common in Civil Service exams.

Mixed Operations (Addition + Multiplication)

Some sequences combine two operations repeatedly.

Example:
2, 5, 15, 45, ___
Pattern: ×3, then +?
Actually: ×3 → 2×3 = 6 but result is 5
So the pattern is: ×3 – 1
Thus: 45 × 3 – 1 = 134

But a more exam-accurate example:
3, 7, 15, 31, 63, ___
Pattern: ×2 + 1
Answer: 63 × 2 + 1 = 127

Alternating Patterns

Every second number follows the same rule.

Example:
10, 7, 9, 6, 8, 5, ___
Odd positions: –1
Even positions: –1
Answer: Next term = 7

Another example:
2, 10, 4, 20, 6, 30, ___
Pattern:

• Odd terms: +2

• Even terms: ×2
  Answer: Even term next → 30 × 2 = 60

Alternating patterns require careful observation.

Square and Cube Series (Power Patterns)

These series use perfect squares or cubes.

Square-based

1, 4, 9, 16, 25, ___
Pattern: n²
Answer: 36

Cube-based

1, 8, 27, 64, 125, ___
Pattern: n³
Answer: 216

Mixed square operations

3, 12, 27, 48, 75, ___
Pattern: n² + 3
(2²+3=7? Not fitting. Let’s try difference)
Differences: 9, 15, 21, 27 → +6
Thus next difference +6 → 75 + 33 = 108

Fibonacci-Type Series

Each number is the sum of the previous two.

Example:
1, 1, 2, 3, 5, 8, ___
Answer: 5 + 8 = 13

Variant:
2, 3, 5, 8, 13, ___
Answer: 8 + 13 = 21

Fibonacci patterns appear occasionally.

Complex or Multi-Step Patterns

Some sequences combine multiple patterns across terms.

Example:
5, 10, 8, 16, 14, 28, ___
Observe:
5 → 10 (×2)
10 → 8 (–2)
8 → 16 (×2)
16 → 14 (–2)
Pattern: ×2, –2
Next term: 14 × 2 = 28

This type is common for trick questions.

How to Quickly Identify the Pattern

In exam settings, speed matters. Use this checklist:

1. Check the difference between numbers (simple or incremental?).

2. Check the ratio (multiplication or division?).

3. Look for alternating behavior (is every other term inconsistent?).

4. Check for patterns in differences (e.g., +2, +4, +6).

5. Consider powers or roots (squares, cubes).

6. Look for mixed operations (×2 + 1).

7. Check if it resembles Fibonacci.

Once you classify the type, solving becomes straightforward.

Sample Civil Service Exam-Style Questions

Below are 20 practice questions with explanations.

Practice Questions (No Answers Yet)

1. 4, 8, 16, 32, ___

2. 5, 12, 19, 26, ___

3. 3, 6, 7, 14, 15, 30, ___

4. 10, 7, 8, 5, 6, 3, ___

5. 2, 3, 6, 18, 72, ___

6. 1, 4, 9, 16, ___

7. 11, 13, 17, 23, ___

8. 100, 90, 81, 73, ___

9. 7, 14, 28, 56, ___

10. 4, 9, 19, 39, ___

11. 20, 25, 35, 50, ___

12. 2, 5, 10, 17, 26, ___

13. 30, 20, 25, 15, 20, ___

14. 1, 2, 3, 5, 8, 13, ___

15. 3, 5, 10, 12, 24, 26, ___

16. 90, 30, 10, 3.33, ___

17. 8, 16, 24, 48, ___

18. 15, 18, 24, 33, ___

19. 6, 12, 21, 33, ___

20. 4, 6, 9, 13, 18, ___

Answer Key and Explanations

1. 64 (×2)

2. 33 (+7)

3. 31 (×2, +1 alternating)

4. 4 (–3, +1 repeating)

5. 360 (×1.5, ×2, ×3, ×4…)

6. 25 (square numbers)

7. 31 (add prime differences: +2, +4, +6, +8…)

8. 66 (–10, –9, –8…)

9. 112 (×2)

10. 79 (+5, +10, +20, +40…)

11. 70 (+5, +10, +15, +20…)

12. 37 (incrementing differences +3, +5, +7, +9…)

13. 25 (–10, +5 repeating)

14. 21 (Fibonacci)

15. 52 (×2, +2 repeating)

16. 1.11 (÷3 repeatedly)

17. 72 (+8, +8, ×2, +24…)

18. 45 (+3, +6, +9, +12…)

19. 48 (+6, +9, +12, +15…)

20. 24 (+2, +3, +4, +5, +6…)

Final Tips for the Number Series Section

• Do not overthink—patterns are usually simple.

• Check both differences and ratios.

• Write down small differences; they reveal the pattern quickly.

• Review common patterns regularly until you recognize them instantly.

• Practice under timed conditions to improve speed.

With mastery of these concepts, number series questions become some of the easiest items in the Civil Service Exam.

Problem Sets: Number Series Practice

Below are practice questions modeled on Civil Service Exam style. Choose the next number (?) in each series.

1. 4, 8, 12, 16, ?

2. 3, 9, 27, 81, ?

3. 10, 7, 4, 1, ?

4. 5, 10, 20, 40, ?

5. 2, 5, 11, 23, 47, ?

6. 7, 14, 21, 28, ?

7. 1, 4, 9, 16, 25, ?

8. 100, 90, 81, 73, ?

9. 2, 3, 5, 8, 12, 17, ?

10. 15, 12, 18, 15, 21, 18, ?

11. 6, 12, 24, 48, ?

12. 5, 11, 23, 47, 95, ?

13. 1, 2, 4, 7, 11, 16, ?

14. 9, 16, 25, 36, 49, ?

15. 80, 70, 63, 57, 52, ?

16. 3, 6, 12, 24, 48, ?

17. 4, 7, 13, 25, 49, ?

18. 2, 6, 18, 54, ?

19. 11, 13, 17, 23, 31, ?

20. 5, 8, 14, 26, 50, ?

21. 30, 20, 25, 15, 20, 10, ?

22. 1, 1, 2, 3, 5, 8, ?

23. 4, 10, 22, 46, 94, ?

24. 7, 9, 13, 21, 37, ?

25. 90, 30, 10, 3.33, ?

Answer Key and Explanations

1. 20

   • Pattern: +4 each time

   • 4, 8, 12, 16, 20

2. 243

   • Pattern: ×3

   • 3, 9, 27, 81, 243

3. –2

   • Pattern: –3 each time

   • 10, 7, 4, 1, –2

4. 80

   • Pattern: ×2

   • 5, 10, 20, 40, 80

5. 95

   • Differences: +3, +6, +12, +24 (each difference ×2)

   • Next difference: 24 × 2 = 48 → 47 + 48 = 95

6. 35

   • Pattern: +7

   • 7, 14, 21, 28, 35

7. 36

   • Perfect squares: 1², 2², 3², 4², 5², 6²

   • Next: 6² = 36

8. 66

   • Differences: –10, –9, –8 → decreasing by 1

   • Next difference: –7 → 73 – 7 = 66

9. 23

   • Differences: +1, +2, +3, +4, +5 …

   • 2, 3, 5, 8, 12, 17, next +6 → 17 + 6 = 23

10. 24

    • Two alternating series:

      • 15, 18, 21, 24 (odd positions: +3)

      • 12, 15, 18 (even positions: +3)

    • Next term is odd position → 21 + 3 = 24

11. 96

    • Pattern: ×2

    • 6, 12, 24, 48, 96

12. 191

    • Differences: +6, +12, +24, +48 (doubling)

    • Next: +96 → 95 + 96 = 191

13. 22

    • Differences: +1, +2, +3, +4, +5 …

    • Next difference: +6 → 16 + 6 = 22

14. 64

    • Squares: 3², 4², 5², 6², 7², 8²

    • Next: 8² = 64

15. 48

    • Differences: –10, –7, –6, –5, …

    • Next difference: –4 → 52 – 4 = 48

16. 96

    • Pattern: ×2

    • 3, 6, 12, 24, 48, 96

17. 97

    • Differences: +3, +6, +12, +24 (doubling)

    • Next difference: +48 → 49 + 48 = 97

18. 162

    • Pattern: ×3

    • 2, 6, 18, 54, 162

19. 41

    • Differences are increasing odd numbers: +2, +4, +6, +8 …

    • Next difference: +10 → 31 + 10 = 41

20. 98

    • Differences: +3, +6, +12, +24 (doubling)

    • Next difference: +48 → 50 + 48 = 98

21. 15

    • Alternating pattern: –10, +5, –10, +5, –10, +5 …

    • 30 → 20 (–10), 20 → 25 (+5), 25 → 15 (–10),
      15 → 20 (+5), 20 → 10 (–10), next: 10 → 15 (+5)

22. 13

    • Fibonacci series: each term = sum of previous two

    • 5 + 8 = 13

23. 190

    • Differences: +6, +12, +24, +48 (doubling)

    • Next difference: +96 → 94 + 96 = 190

24. 69

    • Differences: +2, +4, +8, +16 (doubling)

    • Next difference: +32 → 37 + 32 = 69

25. 1.11

    • Pattern: ÷3 each time

    • 90 ÷ 3 = 30, 30 ÷ 3 = 10, 10 ÷ 3 = 3.33,
      3.33 ÷ 3 ≈ 1.11

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