Sequences

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.

 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 
 

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…

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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.

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…..
 

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.
 

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.

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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.