Most aptitude based examinations ask questions on number series and missing numbers. These types of questions follow a logical approach or pattern based on elementary maths. A random number series is given in which we need to analyse and find out the pattern.

Once the pattern is found out, we need to find the missing number; the missing number can be filled in either at the end of the series or somewhere in between.

A number series consists of a group of numbers following an elementary mathematical formula. There are numerous types of number series, and then we need to learn the concepts involved in each one of them.

Finding the missing number series in an array is fun and easy at the same time if we understand the principles or rules behind it. So, let us try to learn the type of number series and the associated mathematical principles to master the art of finding missing numbers.

**What do you understand by the term ‘Number Sequence’?**

A number sequence is a pattern in a question that helps in unlocking a person’s mathematical aptitude. That is why questions based on number sequence are common in entrance exams and competitive examinations. These mathematical problems revolve around a core principle based on a simple numerical pattern or a logical setup. Thus, a number sequence is an orderly arranged set of numbers representing a numerical pattern based on a logical rule.

Number sequence consists of numbers in progression or an ordered series that builds around a numerical pattern. There are two types of number sequence-

**1) ****Infinite Sequence Numbers**

An infinite number sequence is a number sequence that can continue indefinitely without getting terminated.

**2) ****Finite Sequence Numbers**

A finite number sequence can only continue for a limited range and ends after that.

Any number sequence for it to have a pattern will follow a set of mathematical rules. The examiner produces such a sequence and removes a number in the sequence. There is one number missing in general, but in other highly competitive exams, the missing numbers in the sequence can go up to two numbers.

Therefore, by learning and practising number sequences and sharpening the skillset, a student can easily find the solution. This helps improve the student’s numerical reasoning. It may help in the future with respect to real-life applications like the calculation of taxes, loans, and related fields.

**Missing Number in a Number Series**

To solve missing numbers in number series, we need to understand the different types of number series. Based on these concepts, we can analyse the numerical patterns involved in each sequence. Once we can find the respective pattern for each sequence finding the missing number will be easy.

**Different types of Number Series**

Some of the common mathematical patterns followed by each number series are given below:

**Perfect Square Number Series**

A number series based on perfect squares is one of the easiest and the common types of questions asked in competitive exams.

Example: 121, 144, 169, _, 225?

Answer: √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15.

**Perfect Cube Number Series**

A perfect cube number series is based on the cube root of a number in series or progression. Let us look at the below example for more clarity.

Example: 512, 729, 1000, _

Sol: ∛512 = 8, ∛729 = 9, ∛1000 = 10, ∛1331 = 11

**Geometric Number Series**

Geometric number series can be either based on increasing or decreasing order of numbers. Also, each of these successive numbers is obtained by multiplying the previous number by a specific number.

Generally, a geometric number series in the form of {a, ar, ar², ar³……}

Example: 1,2,4,8, _

Sol: 1, 1×2, 1×2², 1×2³, 1×24=16.

**Triangular Number series**

In triangular number series, the numbers represent a figurative number sequence. Imagine an equilateral triangle that can be formed with the help of balls.

The number of balls keeps on increasing based on the progression in the series, and as such, the size of the triangle increases. The following formula gives a triangular number series-

x n = (n2 n) / 2

Example – 1, 3, 6, 10, 15, _ (Ans. Is 21)

**Arithmetic Number Series**

In arithmetic number series, the following is obtained by adding or subtracting a particular constant number to the previous number.

Example: 4, 9, 14, 19, 24, 29, _

Here, if we subtract the 2nd number with the first, the result is 5. The same is true for the following numbers. So, the constant number which is to be added to get the next number is 5. Therefore, the nest number after 29 is 34.

**Fibonacci Number series**

Fibonacci Number series is a popular mathematical series that forms a numerical pattern in which the next number is obtained by adding the prior first two numbers. An example of the Fibonacci number series is given below –

Example: 0, 1, 1, 2, 3, 5, 8, 13, _

The third number, 1, is obtained by adding 0 and 1. Similarly, the 4th term is obtained by adding 1 and 1. Likewise, the missing number is obtained by adding 8 and 13, which is 21.

**Twin Number series**

A twin number series consists of a combination of two number series. The alternating numbers of a twin series can help in generating a third independent series.

For example, let us assume the twin series – 1, 2, 3, 4, 8, 10, 13, 16.

If we observe it closely, we will find two different series. These two different series are 1, 3, 8, 13 and 2, 4, 10,16.

**Two-Stage Number Series**

In a two-stage number series, the differences of the consecutive numbers create another arithmetic number series.

Example: 1, 3, 6, 10, 15.

Here, (3-1) = 2, (6-3) = 3, (10-6) = 4, (15-10) = 5.

Therefore, we get another arithmetic series, i.e. 2, 3, 4, 5….

**Mixed Number Series**

Mixed Number Series may consist of more than one numerical pattern. This pattern may be arranged in a singular series. Sometimes, it can result in a random numerical pattern.

Example:10, 22, 46, 94, 190, _

Answer:

(10 x 2) = 20 + 2 = 22

(22 x 2) = 44 + 2 = 46

(46 x 2) = 92 + 2 = 94

(94 x 2) = 188 + 2 = 190

(190 x 2) = 380 + 2 = 382.

Therefore, the missing number in this mixed number sequence is 382.

**Conclusion**

For students, questions based on numerical series and mathematical patterns require practice and understanding of core principles. They also need to know the underlying logical reasoning and series formula.

Once students can understand the core principles, solving missing number questions will be a walk in the park. Also, students need to check for the geometrical relationships between the numbers given in the sequence to arrive at the right solution.

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