The **number sequence** is an essential mathematical tool for testing a person’s intelligence. Number series problems are common in most management aptitude exams.

The problems are based on a numerical pattern that is governed by a logical rule. For example, you might be asked to predict the next number in a given series following the laid rule.

* The three prevalent questions in this exam that can be asked are:*

- Identify a term that is wrongly placed in a given series.
- Find the missing number in a certain series.
- Complete a given series.

## What is a Sequence Number?

Number sequence is a progression or an ordered list of numbers governed by a pattern or rule. Numbers in a sequence are called terms. A sequence that continues indefinitely without terminating is an infinite sequence, whereas a sequence with an end is known as a finite sequence.

Logic numerical problems generally consist of one or two missing numbers and 4 or more visible terms.

For this case, a test designer produces a sequence in which the only one fits the number. By learning and excising number sequence, an individual can sharpen their numerical reasoning capability, which helps our daily activities such as calculating taxes, loans, or doing business. For this case, it is important to learn and practice number sequence.

*Example 1*

*Which list of numbers makes a sequence?*

- 6, 3, 10, 14, 15, _ _ _ _ _ _
- 4,7, 10, 13, _ _ _ _ _ _

__Solution__

The first list of numbers does not make a sequence because the numbers lack proper order or pattern.

The other list is a sequence because there is a proper order of obtaining the preceding number. The consecutive number is obtained by adding 3 to the preceding integer.

*Example 2*

*Find the missing terms in the following sequence:*

8, _, 16, _, 24, 28, 32

__Solution__

Three consecutive numbers, 24, 28, and 32, are examined to find this sequence pattern, and the rule obtained. You can notice that the corresponding number is obtained by adding 4 to the preceding number.

The missing terms are therefore: 8 + 4 = 12 and 16 + 4 = 20

*Example 3*

*What is the value ofnin the following number sequence?*

12, 20,*n*, 36, 44,

__Solution__

Identify the pattern of the sequence by finding the difference between two consecutive terms.

44 – 36 = 8 and 20 – 12 = 8.

The pattern of the sequence is, therefore, the addition of 8 to the preceding term.

So,

*n*= 20 + 8 = 28.

## What are the Types of Number Sequence?

There are many number sequences, but the arithmetic sequence and geometric sequence are the most commonly used ones. Let’s see them one by one.

### Arithmetic Sequence

This is a type of number sequence where the next term is found by adding a constant value to its predecessor. When the first term, denoted as x_{1}, and d is the common difference between two consecutive terms, the sequence is generalized in the following formula:

x_{n }= x_{1 }+ (n-1) d

where;

x_{n }is the n^{th }term

x_{1} is the first term, n is the number of terms and d is the common difference between two consecutive terms.

*Example 4*

By taking an example of the number sequence: 3, 8, 13, 18, 23, 28……

The common difference is found as 8 – 3 = 5;

The first term is 3. For instance, to find the 5^{th} term using the arithmetic formula; Substitute the values of the first term as 3, common difference as 5, and the n=5

5^{th} term =3 + (5-1) 5

=23

*Example 5*

It important to note that the common difference is not necessarily a positive number. There can be a negative common difference as illustrated in the number series below:

25, 23, 21, 19, 17, 15…….

The common difference, in this case, is -2. We can use the arithmetic formula to find any term in the series. For example, to get the 4^{th }term.

4^{th} term =25 + (4-1) – 2

=25 – 6

=19

### Geometric Series

The geometric series is a number series where the following or next number is obtained by multiplying the previous number by constant known as the common ratio. The geometric number series is generalized in the formula:

x_{n }= x_{1 }× r^{n-1}

where;

x _{n} = n^{th }term,

x_{1 }= the first term,

r =common ratio, and

n = number of terms.

*Example 6*

For example, given a sequence like 2, 4, 8, 16, 32, 64, 128, …, the n^{th} term can be calculated by applying the geometric formula.

To calculate the 7^{th} term, identify the first as 2, common ratio as 2 and n = 7.

7^{th} term = 2 x 2^{7-1}

= 2 x 2^{6}

= 2 x 64

= 128

*Example 7*

A geometric series can consist of decreasing terms, as shown in the following example:

2187, 729, 243, 81,

In this case, the common ratio is found by dividing the predecessor term with the next term. This series has a common ratio of 3.

### Triangular series

This is a number series in which the first term represents the terms linked to dots presented in the figure. For a triangular number, the dot shows the amount of dot required to fill a triangle. Triangular number series is given by;

x n= (n^{2}+ n) / 2.

*Example 8*

Take an example of the following triangular series:

1, 3, 6, 10, 15, 21………….

This pattern is generated from dots that fill a triangle. It is possible to get a sequence by adding dots in another row and counting all the dots.

### Square series

A square number is simplifying the product of an integer with itself. Square numbers are always positive; the formula represents a square number of series

x_{ n }= n^{2}

*Example 9*

Take a look at the square number series; 4, 9, 16, 25, 36………. This sequence repeats itself by squaring the following integers: 2, 3, 4, 5, 6…….

### Cube series

Cube number series is a series generated by the multiplication of a number 3 times by itself. The general formula for cube number series is:

x_{ n} = n^{3}

### Fibonacci series

A mathematical series consists of a pattern in which the next term is obtained by adding the two terms in-front.

*Example 10*

An example of the Fibonacci number series is:

0, 1, 1, 2, 3, 5, 8, 13, …

For instance, the third term of this series is calculated as 0+1+1=2. Similarly, the 7^{th} term is calculated as 8 + 5 = 13.

### Twin series

By definition, a twin number series comprises a combination of two series. The alternating terms of twin series can generate another independent series.

An example of the twin series is 3, 4, 8, 10.13, 16, …..By closely examining this series, two series are generated as 1, 3, 8,13 and 2, 4, 10,16.

### Arithmetico-Geometric Sequence

This is a series formed by the combination of both arithmetic and geometric series. The difference of consecutive terms in this type of series generates a geometric series. Take an example of this arithmetico -geometric sequence:

1, 2, 6, 36, 44, 440, …

### Mixed Series

This type of series is a series generated without a proper rule.

*Example 11*

For example; 10, 22, 46, 94, 190, …., can be solved using the following steps:

10 x 2= 20 + 2 = 22

22 x 2 = 44 + 2 = 46

46 x 2 = 92 + 2 = 94

190 x 2 = 380 + 2 = 382

The missing term is therefore 382.

## Number pattern

Number pattern is generally a sequence or a pattern in a series of terms. For example, the number pattern in the following series is +5:

0, 5, 10, 15, 20, 25, 30………

In-order to solve number pattern problems, closely check the rule governing the pattern.

Try by addition, subtraction, multiplication, or division between consecutive terms.

## Conclusion

In summary, problems involving number series and pattern require checking the relationship between these numbers. You should check for an arithmetic relationship such as subtraction and addition. Check for geometric relationships by dividing and multiplying the terms to find their common ratio.

## FAQs

### What is sequence and example? ›

A sequence is **a list of numbers in a certain order**. Each number in a sequence is called a term . Each term in a sequence has a position (first, second, third and so on). For example, consider the sequence {5,15,25,35,…} In the sequence, each number is called a term.

**What are the 4 types of sequences? ›**

There are four main types of different sequences you need to know, they are **arithmetic sequences, geometric sequences, quadratic sequences and special sequences**.

**What is the rule for number sequence? ›**

Number sequences are sets of numbers that follow a pattern or a rule. **If the rule is to add or subtract a number each time, it is called an arithmetic sequence**. If the rule is to multiply or divide by a number each time, it is called a geometric sequence. Each number in a sequence is called a term.

**How do you explain a sequence to a child? ›**

Kids easily learn that one thing follows another. Their routines at home provide great examples, and are a good introduction to the concept of sequencing. For example, **first we eat dinner, then we take a bath, after that we read stories, and finally we turn out the light.**

**How do you explain sequencing to a child? ›**

Ask the child questions about what happened first, next, and last. Then, mix up the pictures and place them back in front of the child. Ask the child to pick out which step happened first and place that on a sequencing strip (like in the picture above) or just in front of them on the table.

**How do you create a sequence explain with example? ›**

Example 1:

**CREATE SEQUENCE sequence_1 start with 1 increment by 1 minvalue 0 maxvalue 100 cycle;** Above query will create a sequence named sequence_1. Sequence will start from 1 and will be incremented by 1 having maximum value 100. Sequence will repeat itself from start value after exceeding 100.

**What is sequence order example? ›**

A sequence is a set of numbers in a particular order or a set of numbers that follow a pattern. The most basic sequential order example is that of **counting numbers 1, 2, 3, 4 and so on**. The numbers follow an increasing pattern (1, 2, 3…).

**What are the two examples of sequence? ›**

**Some of the most common examples of sequences are:**

- Arithmetic Sequences.
- Geometric Sequences.
- Harmonic Sequences.
- Fibonacci Numbers.

**What type of sequence is 1 1 2 3 5 8? ›**

The **Fibonacci sequence** of whole numbers is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,... The sequence is widely known for its many intriguing properties.

**What is the formula of sequence? ›**

An arithmetic sequence can be defined by an explicit formula in which **a _{n} = d (n - 1) + c**, where d is the common difference between consecutive terms, and c = a

_{1}.

### How do I teach my child the number sequence? ›

In teaching number sequences, it is important to **start with concrete examples using blocks or other manipulatives**. It is easiest to start by showing the growth of a simple repeating pattern. Simple repeating pattern. Show how it grows by adding successive identical units of repeat.

**Can I define sequence in my own understanding? ›**

A sequence of things is **a number of them that come one after another in a particular order**.

**What is a sequence simple definition? ›**

sequence. / (ˈsiːkwəns) / noun. an arrangement of two or more things in a successive order. the successive order of two or more thingschronological sequence.

**How do you introduce a number sequence? ›**

**Top tips for teaching number sequences**

- Teach them rhymes and games.
- Incorporate numbers into daily tasks.
- Patterns don't have to be numbers.

**What are the 4 principles of sequencing? ›**

The principles of sequencing content described by Print (1993 as cited in Edith Cowan University, 2001) are: **Simple to complex, prerequisite learning, whole to part, and chronology**. These four principles have become increasingly acceptable as the criteria for sequencing contents.

**What are the three types of sequencing? ›**

**Key Sequencing Methods**

- DNA Sequencing. Analyze the entire genome, focus on regions of interest with whole-exome and targeted sequencing, or study DNA-protein interactions.
- RNA Sequencing. ...
- Methylation Sequencing. ...
- High-Throughput Sequencing.

**What is an example of a sequence in real life? ›**

Suppose I'm eating a box of cookies. At the first time I eat 1 cookie, and ½ at the second time. And every time afterwards, I'll eat half amount of the previous. **The amount of cookies I eat every time** is a sequence made by a list of numbers.

**What are number patterns examples? ›**

Few examples of numerical patterns are: **Even numbers pattern -: 2, 4, 6, 8, 10, 1, 14, 16, 18**, … Odd numbers pattern -: 3, 5, 7, 9, 11, 13, 15, 17, 19, … Fibonacci numbers pattern -: 1, 1, 2, 3, 5, 8 ,13, 21, … and so on.

**Which method is used to solve sequencing? ›**

**SAI method** is used to frame a sequence of jobs for processing the n -jobs on m -machines in such a way that the total elapsed time is minimized.

**What is the next number in the sequence 2 3 4 6 6 9 8 12 10? ›**

Detailed Solution

Hence, the correct answer is **15**.

### What is the sequence of 1 4 9 16? ›

Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, **0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144**, and so on, are all square numbers.

**What is the pattern of 2 3 5 7 11 13? ›**

The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. The sequence of gaps—the differences between each prime and the next—is **1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2 and 4**.

**What are all the sequences in math? ›**

There are mainly four types of sequences in **Arithmetic, Arithmetic Sequence, Geometric Sequence, Harmonic Sequence, and Fibonacci Sequence**.

**How do you teach number sequences? ›**

Step 1 – Number sequence using objects

**Start by displaying five uniform objects horizontally.** Demonstrate how to match one number to each object, one through five. Say each number as you place it in sequence. Then, have students match each number, in order, to the objects (our program uses bus stops).

**How do you teach number sequences to children? ›**

**Encourage them to use their fingers to count forwards and backwards**, so they reinforce their number learning. As children get used to numbers and their confidence using them grows, our Sequence Snakes! Worksheets are a wonderful way to get them used to finding numbers with common relationships.

**What is sequence in simple words? ›**

**a series of related things or events, or the order in which they follow each other**: The first chapter describes the strange sequence of events that led to his death.

**What is the purpose of number sequence? ›**

Number sequences are used **to generate readable, unique identifiers for master data records and transaction records that require identifiers**. A master data record or transaction record that requires an identifier is referred to as a reference.

**How do you answer number sequence questions? ›**

It is based on either descending or ascending order of numbers and **each successive number is obtained by dividing or multiplying the previous number by a specific number**. Example:4, 36, 324, 2916? Sol:4 x 9 = 36, 36 x 9 = 324, 324 x 9 = 2916, 2916 x 9 = 26244.

**What is the simplest sequence of numbers that children learn? ›**

**Rote counting** is the simplest number concept that children can develop. It is the ability to say numbers in order from memory, so it basically consists of counting numbers sequentially. Most children find it easy to count by rote.

**What are the key words for sequence? ›**

It has been designed to give students a basic idea of keywords that they may see in text and to help them identify the sequence of events in a story; using the major key words **First, Next, Then, Last, and Finally**.