Number Series

A number series is an ordered sequence of numbers arranged according to a specific mathematical rule or pattern. Each number in the series relates to the others through consistent operations like addition, subtraction, multiplication, division, exponents, or more complex relationships.

Number Series is a widely asked topic in the Logical Reasoning section of competitive examinations. There are two main types of questions asked,

1. Missing Term Series: A sequence is given with one or more missing numbers, and you must find the correct number that fits the pattern.

2. Wrong Term Series: A complete sequence is given, but one number is incorrect, and you must identify the wrong term and sometimes correct it.

Types of Number Series

Number series can be categorized based on their underlying patterns. Here are the most common types:

1. Arithmetic Series

Rule: Each term increases or decreases by a fixed difference (common difference).

Example:

• 5, 9, 13, 17, ... (Here number is increased by +4)
• 20, 17, 14, 11, ... (Here number is decreased by −3)

2. Geometric Series

Rule: Each term is multiplied/divided by a fixed ratio (common ratio).

Example:

• 3, 6, 12, 24, ... (Here number is multiply ×2)
• 81, 27, 9, 3, ... (Here number is divided by 3)

3. Square Series

Rule: Terms are squares of natural numbers().

Example:

1, 4, 9, 16, 25, ... (1², 2², 3², ...)

4. Cube Series

Rule: Terms are cubes of natural numbers ).

Example:

 1, 8, 27, 64, 125, ... (1³, 2³, 3³, ...)

5. Fibonacci Series

Rule: Each term is the sum of the two preceding terms.

Example:

 0, 1, 1, 2, 3, 5, 8, 13, ... 

6. Prime Number Series

Rule: Consists of prime numbers (divisible only by 1 and themselves).

Example:

2, 3, 5, 7, 11, 13, 17, ...

7. Alternating Series

Rule: Combines two or more different operations alternately.

Examples:

• 1, 4, 3, 6, 5, 8, ... (Alternate between +3 and −1)
• 2, 6, 12, 20, 30, ... (1×2, 2×3, 3×4, 4×5, ...)

 8. Factorial Series

Rule: Terms are factorials ().

Example:

 1, 2, 6, 24, 120, ... (1!, 2!, 3!, 4!, ...)

9. Mixed Series

Rule: Combines multiple arithmetic/geometric/other operations.

Examples:

• 2, 5, 10, 17, 26, ... (n² + 1: 1²+1=2, 2²+1=5, etc.)
• 1, 3, 7, 15, 31, ... (×2 +1: 1×2+1=3, 3×2+1=7, etc.)

10. Exponential/Power Series

Rule: Terms involve exponents or power functions.

Examples:

2, 4, 16, 256, ... (Each term is the square of the previous one: 2²=4, 4²=16, etc.)

3, 9, 81, 6561, ... (3¹, 3², 3⁴, 3⁸, ... exponents double each time)

Important Patterns of Number Series:

1. Series with an increasing difference (most commonly asked)
2. Series with a constant difference (the difference will be the same) 
3. Series with decreasing difference (the difference will be decreasing)
4. Perfect squares and cubes of numbers’ series (concept of squares and cubes will be there, either directly or indirectly) 
5. Miscellaneous series(which may consist of different operations together, like  addition, subtraction, division, etc)

Tips and Tricks to Solve Number Series Questions:

1. The Easiest way to approach number series questions is to observe the difference between the various  terms. Here are some methods and tips you can use to solve number series questions:
2. If you observe carefully a constant difference between the different numbers, it means that the question belongs to the series with a constant degree category. 
3. If you observed carefully the difference between the various numbers it is either increasing or decreasing, then the question belongs to either the series with an increasing difference or the series with decreasing difference respectively. 
4. In case, you are not able to spot an increasing or decreasing difference between the numbers, try to divide the 2nd term of the series with the first, the 2nd term with the 3rd term and so on. If the answer to the constant division comes as the same number, then this question belongs to the product series. 
5. In case, none of the above approach works, you can write every term of the question as to the multiplication of 2 factors and try to spot a pattern between the terms. 
6. If you are not able to spot a pattern and the difference between the terms is decreasing or increasing at an accelerated rate, you can try for the square/cube series. 

Also Check:

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