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Last updated on July 9th, 2025

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Sequences

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Sequences are mathematical concepts with many real-life applications. They follow a specific pattern. Let’s learn more about sequences through their different perspectives.

Sequences for Australian Students
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What is a Sequence?

A sequence is a list of ordered numbers and signs arranged in a particular pattern. There are many types of sequences, like finite or infinite. The elements in a sequence are called terms. If the sequence consists of specific numbers and comes to an end, then that particular order is called finite. If the sequence does not have an end, then it is called an infinite sequence. To compute each term in a sequence, we have an explicit formula. Sequences can be classified into many sections according to their types. Here are some examples of different sequences. Some common ones include arithmetic sequence, triangular sequence, quadratic sequence, geometric sequence, etc. Let’s learn more about sequences by discovering their various aspects.

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What is the Order of Sequence?

 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 is descending from a large number to a small number, and 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 
 

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What are The Types of Sequences?

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 increases or decreases by a fixed common difference. 
Example:
A). 2,4,6,8,....... In this sequence, the numbers are increasing by adding 2; here d = 2.
B). 8,6,4,2……. In this sequence, the numbers are decreasing. Here, the difference is -2.
 
We can find this sequence through Equations:
Here is a sequence:
     3,6,9,12……..
a = First term
d  = common difference
n  = Term number
an= nth term
First term a = 3
Common difference d  = 6 – 3 = 3
 The general formula for finding a sequence is = an= a + (n-1) d 
Step 1.  We should put the values in the formula.
 a = 3, d = 3.  Formula :  an= a + (n-1) d
an  = 3 + 3n - 3
an   = 3n
Therefore, an = 3n  general formula of this sequence.
 Let’s find the actual sequence by using an = 3n:                                
3(1) = 3, 3(2) = 6, 3(3) = 9, 3(4) = 12
In short, the sequence: 3, 6, 9, 12…

 

.
Quadratic Sequence


This 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. This type of sequence is called a quadratic sequence.
To make it clearer, let's use some examples:
Sequence: 1,4,9,16,25……
Step 1. Find the common difference : d = n-1
d = 4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9. Here, the first differences are not equal, but we get another sequence by combining all the above differences. The sequence is 3, 5, 7, 9,...
Let's check the difference of this sequence. d = n-1
 5 - 3 = 2
 7 - 5 = 2
 9 - 7 = 2
 Look here, the second level difference is the same. Such sequences are quadratic sequences.

 


Geometric Sequence


In geometric sequences, there should be a common ratio between all the terms. General Formula of geometric sequence: a, ar, ar2, ar3, ar4, ……..
a = first term
r = ratio
 Example:
2,6,18,54,162…
Here, a = 2 
         r =  3
It can be written in : 2, 23 = 6, 63 = 18, 183 = 54,...
These are the important aspects of a geometric sequence as discussed in this heading.

 


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) = 15 1+52n-1-52n Here, F(n) is the nth term of this sequence.
Example: Let’s start with F(0)=0, F(1)=1.

                 

n F(n)
0 0
1 1
2 0+1=1
3 1+1=2


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, 1/2, 1/3, 1/4...... 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…..
 

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What are the Formulas of Sequences?

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

 


1. Arithmetic sequence, Tn (nth term)= a + (n-1)d
a= first term, d= common difference

 

 

2. Geometric sequence an = arn-1
a = first term
r = common ratio

 

 

3. Fibonacci sequence =  F(n - 1) + F(n - 2)

 

 

4. Square number sequence an = n2

 

 

5. Cube number sequence an = n3

 

 

6. Triangular number sequence Tn = n(n + 1 )2  

 

 

What are the Rules of Sequence?


 Sequences mainly follow two types of rules. Each rule has its specific features.
Two rules: 
a. Implicit: in this rule, one term is defined by using its previous term that comes before it.
B. 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
an= an-1+2
Explicit an = a+(n+1) d    
                 =  3 =2n-2
                  = 2n+1

 

 

How to find the Missing Number in Sequence?


 We can find the missing numbers of a sequence by using a general formula. For that, we should analyze the pattern of the sequence first, then apply the formula that is appropriate. But for some time, the given sequence didn’t have any specific pattern or 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.
 

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Real-Life Applications of Sequence.

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. 

 


1. 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.

 


2. 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.

 


3. The connection between the Fibonacci sequence to nature.


Yes, the Fibonacci sequence can be seen in the scales of pine cones, the number of petals on the flower, the spiral shape of shells.

 


4. 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.

 


5. 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. 
 

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Common Mistakes and How to Avoid Them in Sequence.

While learning sequences, people may have many confusions and make mistakes. Let’s discuss some common mistakes that people make while solving problems. 

Mistake 1

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Mistakenly analyze an arithmetic sequence as a geometric sequence.
 

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3,6,9,12…..
Here, the sequence is going in a + 3 format. But the student observed this as a geometric sequence. To solve this, we should understand the sequence. 
The sequence is going in a -3 format. Like this: 6-3 = 3, 9-6 = 3, 12-9 =3
The common difference (d)= 3, a constant value, is kept between each term.
 

Mistake 2

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Wrong use of the formula to find nth term.
 

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 In an arithmetic sequence, we can subtract 1 from n. The formula is
an = a + (n-1) d. We should thoroughly understand each formula and the rules while learning.
 

Mistake 3

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 False assumption of sequence without checking.
 

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This happens when we don’t observe the sequence. We should observe the sequence to see whether it starts from 0 or 1.
Example: 2,4,6,8,......
a = 0 it is a false interpretation.
Here, a = 2 (the first term) is correct.
 

Mistake 4

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Misunderstanding of signs or errors.
 

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-4, -8, -12…..
Difference(d) = 4.  Is wrong wrong. To solve this, we need to go through it.
 -8 - (-4) = -4. So the answer is -4. Be careful while observing signs.
 

Mistake 5

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Taking shortcuts to reach an answer.

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Skipping steps in math is not always possible, when we go through implicit rules, it only produces 1 term at a time. 
Solution: Let's go through an example:
Example: Rule a1 = 2, an = an-1 + 5
Find a4? 
Incorrect shortcut = an = 2 + 5 = 7
Correct solution: a1 = 2
a2 = 2 + 5 = 7
a3 = 7 + 5 = 12
a4 = 12 + 5 =17  
This is the correct step we need to follow.
 

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Solved Examples of Sequences.

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Problem 1

Henry, a passenger, got on a bus. The bus charges $4 for a mile, then it adds 1·5 for each mile. How much did it cost him cost to travel 20 miles?

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4, 5·50, 7, 8.50, 10……32·5
 

Explanation

a = 4
Common difference (d) = 1·5
Here we need to use the formula to find the nth term.
Tn (nth term) = a + (n-1) ·d
We need to find the 20th term (cost to pay for 20 miles)
= 4+ (20-1) · 1· 5
a20  = 4+ 19· 1·5
 = 4 + 28·5 
 = 32·5
So the sequence is going like:  4, 5·50, 7, 8.50, 10……32·5
 

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Problem 2

Melon runs a flower shop. She plants flowers in an order of 1,4,9,..... 1 flower in row 1, one like that. Then, on which row did she plant 200 flowers?

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20
 

Explanation

Observing the given sequence, we understand that this is a square number sequence. The formula of this sequence is an = n2  .  
200 = n2
   n  =  20

Therefore, she planted 200 flowers in the 20th row.
 

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Problem 3

In a harmonic sequence, the 6th and 11th terms are 10 and 18. Find its common difference from the given sequence?

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- 2/225
 

Explanation

A harmonic sequence is a sequence in which the reciprocals of the terms form an arithmetic sequence.
So for harmonic sequence a1, a2, a3 . . .
The arithmetic sequence is 1a1, 1a2,1a3. . . 
Let the terms of the harmonic sequence be
a6 = 10 1a6=110
A11 = 18 1a11=118
The formula for nth term in arithmetic sequence is an = a + (n - 1)d
a = 1a1, is the first term of the reciprocal sequence.
D = common difference of the reciprocal sequence.
a + 5d = 110 (for 6th term)
a + 10d = 118 (for 11th term)
Subtract the equations to eliminate a
(a+10d)-(a+5d)=118-110
5d = 118-110
Calculating RHS,
118-110=10-18180=-8180=-245
5d = -245
d = - 2225
 

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Problem 4

How to find the 8th term of the given arithmetic sequence : 5,9,13,17……

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33
 

Explanation

a (first term) = 5 
d = 4 
The formula for solving this problem is an = a + (n-1)d
Applying the value a8  = 5 + (8-1)·4
 = 5 + 28 = 33
 

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Problem 5

How to find the general formula (nth term) of a sequence. -2, 1,4,7……

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an = 3n-5
 

Explanation

 First term a = -2, d = 3 
Applying the formula 
= an = a+(n-1)d
= -2 + (n-1)·d
= -2+ 3n -3 = 3n -5
 

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FAQ’s on sequences

1.Do sequences always have single numbers?

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2. Which formula is used to find the sum arithmetic sequence?

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3.How to identify an arithmetic sequence and a geometric sequence?

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4.The difference between sequence and series.

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5.How many types of sequences are there?

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6.How does learning Algebra help students in Australia make better decisions in daily life?

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7.How can cultural or local activities in Australia support learning Algebra topics such as Sequences?

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8.How do technology and digital tools in Australia support learning Algebra and Sequences?

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9.Does learning Algebra support future career opportunities for students in Australia?

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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