A sequence is an ordered list of numbers (or elements) that follows a specific pattern or rule. Each number in the list is called a term. Sequences can be classified as finite, meaning they have a fixed number of terms, or infinite, meaning they continue without end.
 

There are many types of sequences in mathematics, each defined by its own rule. Common examples include arithmetic, geometric, quadratic, triangular, square, and cube number sequences.

Examples of sequences are: 2, 4, 6, 8, 10…, 6, 12, 24…, 1, 3, 6, 10…

In mathematics, sequences and series are closely related but not the same. Here are the differences between sequence and series. 
 

 There are many types of sequences, as we mentioned above. A sequence can have two types of order. They are: 

• Ascending Order 
  
   
• Descending Order

Let’s go through one by one:

Ascending Order: When the order is in increasing pattern, it is called ascending order.

Examples of ascending order.
3,6,9,12,15,18,21……. d = 3
4,8,12,16,20,24,28….. d = 4
5,10,15,20,25,30……. d = 5

The above three sequences had a constant value. They are increasing in a specific pattern. Therefore, we can see that these sequences are multiplied in an exact pattern.

Descending Order: If a sequence decreases from a large number to a small number, it is called descending order.

Let’s check how it is:
10,9,8,7,6,5…….. This sequence is decreasing by 1
15,12,8,6,4…….. This sequence is decreasing by 4
42,35,28,21……. This sequence is decreasing by 7

So these are some examples of descending order 

In mathematics, sequences are classified into finite and infinite sequences. The classification is based on the number of terms they contain.

Finite Sequence: A finite sequence is a sequence that contains a limited number of terms. It starts at a particular term and ends at a definite point, so all of its terms can be counted.

Examples: 

• The sequence of the first five natural numbers: 1,2,3,4,5
   
• The sequence of the first six multiples of 5: 5, 10, 15, 20, 25, 30

Infinite Sequence: An infinite sequence has an unlimited number of terms. It lacks a final term and continues indefinitely.

Examples: 

• The sequence of natural numbers: 1,2,3,4,5,6, … 
   
• The sequence of odd numbers: 1, 3, 5, 7, 9, …

There are two types of sequences, Finite sequences and infinite sequences. Along with that, there are : 

• Arithmetic Sequence
• Quadratic Sequence
• Geometric Sequence
• Fibonacci Sequence
• Harmonic Sequence
• Triangular Sequence
• Square Number Sequence
• Cube Number Sequence

These 8 sequences are important. We should be aware of these 8 sequences.

Arithmetic Sequence:  An arithmetic sequence is a sequence of numbers in which each term is obtained by adding or subtracting a fixed number from the previous term. This fixed number is called the common difference (d). The common difference can be positive, negative, or zero. For example, 2,4,6,8,... where d is 2. 

Where, 
a = first term
d = common difference

The formula to find the nth term of an arithmetic sequence is: a_n = a + (n - 1)d.

Where, 
a_n is the nth term
 
n is the term number 

Let’s find the nth term of the sequence: 3, 6, 9, 12, … 

Here, a = 3

d = 6 - 3 = 3

a_n = a + (n - 1)d

a_n = 3 + (n - 1)3

a_n = 3 + 3n - 3

So, a_n = 3n


Quadratic Sequence: The quadratic sequence is yet different from the arithmetic sequence. In an arithmetic sequence, there is a fixed constant value.  In this sequence, the first difference between the terms is not the same; instead, the second difference of the sequence is the same.

Example: Consider the sequence: 1, 4, 9, 16, 25, … 

Finding the first difference by subtracting each term from the one that follows: 

• 4 - 1 = 3
   
• 9 - 4 = 5
   
• 16 - 9 = 7
   
• 25 - 16 = 9

The first differences from a new sequence: 3, 5, 7, 9, … 

As these differences are not equal, the sequence is not arithmetic. 

Finding the second term by finding the difference of the first differences: 

• 5 - 3 = 2
   
• 7 - 5= 2
   
• 9 - 7 = 2

The second difference applies to all. 

Since the second differences are constant, the given sequence is quadratic.

Geometric Sequence: A geometric sequence is a sequence in which each term is obtained by multiplying or dividing the previous term by a fixed number called the common ratio (r). The ratio between any two consecutive terms in a geometric sequence remains constant.

The general form of a geometric sequence is: a, ar, ar², ar³, ar⁴, ……..

Where, 

a is the first term 

r is the common ratio

The formula to find the nth term of a geometric sequence is: a_n = ar^{n - 1}
 
For example, consider the sequence 2, 6, 18, 54, 162,.. 

Here, a = 2

r = 3

Here, a_n = 2 × 3^{n - 1}

Fibonacci Sequence: In this sequence, there is a series of numbers, each term in the series is the sum of the two preceding ones. The general formula of this sequence is called a closed formula, also known as Binet’s formula.
\(F(n) = \frac1{√5} [( \frac{1 + \sqrt5}{2}) n - (\frac{1 -\sqrt5}{2} )n ]\) is the nth term of this sequence.

Example: Let’s start with F(0)=0, F(1)=1.

So the sequence is 0,1,2,3,5,8….

Harmonic sequence: A harmonic sequence is formed by taking the reciprocal of an arithmetic sequence.

Example: 1, \({1\over2}, {1\over3}, {1\over4}...... \) this sequence should be a harmonic sequence: 1,2,3,4,5…with d = 1.

Triangular Number Sequence: This sequence follows a pattern that forms an equilateral triangle.

Example: 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10  

So the sequence is like 1,3,6,10…..

Square Number Sequence: This sequence builds a series of numbers to create a square. To make it clearer by using examples.

Example: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16

 = 1,4,9,16…..

Cube Number Sequence: The sequence that forms cubes by using a series of numbers is called the cube number sequence.

Example:
1³ = 1
2³ = 8
3³=27

The sequence should be 1,8,27…..
 

Every sequence has its formulas, and we discussed them in the above section. Here we are dealing with the formula of n^th term of each sequence.
The formula of nth of each sequence is as follows:

• 1. Arithmetic sequence, T_n (nth term)= a + (n-1)d
  
  a= first term, d= common difference
   
•  Geometric sequence a_n = ar^n-1
  a = first term
  r = common ratio
   
• Fibonacci sequence =  F(n - 1) + F(n - 2)
   
•  Square number sequence a_n = n²
   
•  Cube number sequence a_n = n³
   
•  Triangular number sequence T_n =  \(\frac{n (n +1)}{2}\)
   

Sequences mainly follow two types of rules. Each rule has its specific features. Two rules: 

• Implicit: In this rule, one term is defined by using its previous term that comes before it.
   
• Explicit: Each term is defined directly using a general formula.

Example:
Implicit: 3,5,7….. This is an odd sequence.

a = 3
 
d = 5 - 3 = 2

a_n= a_n-1+2

Explicit a_n = a+(n+1) d    

                 =  3 =2n-2

                  = 2n+1

We can find the missing numbers of a sequence by using a general formula.First, analyze the pattern of the sequence, then apply the appropriate formula. Sometimes, the given sequence may not follow a standard formula. In such cases, we can find the missing terms.

Example:  Find the missing numbers?
2,12,36,80…

Let’s observe the sequence
2 = 1²+1³
12 = 2²+2³
36 = 3²+3³
80 = 4² + 4³

The upcoming number should be 5² + 5³ = 25 + 125 = 150.

We find the missing number of a sequence by using this method.

Learning sequences help students understand number patterns and serve as a foundation for learning sequences and series equations. Here are a few tips and tricks to master sequences. 

• Understand sequences first by identifying their patterns.
   
• Teachers can introduce the sequence of formulas step by step, explaining where each formula comes from rather than just memorizing them.
   
• Teachers can use visual aids, such as Tables, charts, and number lines, to help explain sequences in math.
   
• Understand the difference between a sequence and a series. A sequence is a list of numbers arranged in order, whereas a series is the sum of a sequence. 
   
• Parents can use fun activities like board games and pattern puzzles to make learning sequences enjoyable.

Sequences have many real-life applications. Like, in the fields of finance, construction, population growth, and nature. The arithmetic sequence pattern is followed in finance and construction. 

• Application of sequence in the field of finance: Sequences are mainly used in the savings and investment sector of banking. When we deposit a fixed amount in the bank, it starts to grow in a pattern of arithmetic sequence. The sequence increases by a fixed value. Example: Imagine you deposit 1000/- initially, it starts to increase like 1000, 2000,3000…. Here (d = 1000) here. Savings often follow a geometric sequence due to interest.
   
• The role of sequence in population growth: The population growth depends on sequence. Geometric sequence is related to this field because it has a constant rate. Example: Consider population growth is doubled every year. It grows like 3000, 6000, 9000, 12000,.... Here the sequence is multiplied by 3, showing the growth rate.
   
• The connection between the Fibonacci sequence to nature: The Fibonacci sequence can be seen in the scales of pine cones, the number of petals on the flower, the spiral shape of shells.
   
• The role of sequence in the field of construction is crucial: An arithmetic sequence is applied in the field of construction. Staircase steps and stadium seats are arranged in a fixed pattern. For example a stair has been built with a height of 6 inches, then it increases in such a pattern.  An arithmetic sequence has a fixed value, or it follows a constant value.
   
• Relation between medicines and sequences: In medicine, sometimes dosages are increased by a fixed amount. Arithmetic sequences are used in the medical field. In that case, a sequence is produced there. For example, 50g, 100g, 150g…… Here (d = 50), a constant value is fixed.