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# What is a sequence? Define its all types through examples

- April 14, 2018
- Posted by: allexamshelps
- Category: Mathematics

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### Solution:

#### The number of persons standing in a queue forms a sequence because each of them has a particular order. That is, the first person has the first order, the second one has the second order and so on. when the objects, persons etc. have a particular discernable order, that list of numbers, persons or objects etc., form a sequence.

**Finite sequence:**

#### When we have the last term, it is a finite sequence:

#### Ex: 3, 8, 13, 18…23 where the last term is 23.

### Infinite sequence: \frac { 20 }{ 6 }=6, 6.6, 6.66. 6.666, 6.666…

### The sequence of even natural numbers:

#### Even numbers are written as { a }_{ 1 }=2n

#### Then, the value of n is taken from the list of natural numbers starting from 1 up to n.

#### That is, if n=1 then 2n=2x1=2 or

#### { a }_{ 1 }=2\times 1=2

#### If n=2, then

#### { a }_{ 2 }=2\times 2 = 4

#### If n=3 then

#### { a }_{ 3 }=2\times 3 =6

#### If n=4 then

#### { a }_{ 1 }=2\times 4=8

#### Hence { a }_{ n }=2n gives us even numbers which are divisible by 2.

**The sequence of Odd numbers is written as 2n-1**

#### Then, the value of n is taken from the list of natural numbers starting from 1 up to n.

#### That is, if n=1 then { a }_{ 1 }=2\times 1-1=1

#### If n=2, then { a }_{ 2 }=2\times 2-1=3

#### If n=3 then { a }_{ 3 }=2\times 3-1=5

#### If n=4 then { a }_{ 4 }=2\times 4-1=7

#### Hence { a }_{ n }=2n-1 gives us even numbers which are not divisible by 2.

**1 is an odd number.**

**Fibonacci sequence:**

#### 2, 2, 4, 6, 10, 16…

#### Now clear pattern can be seen above, but if looked carefully it can be detected that third number

#### 4 is the summation of the first two numbers i.e. 2+2=4, and the fourth number is the summation

#### of the second and the third number i.e. 2+4=6 and so on.

#### The third number= the summation of two preceding terms

#### 6= 4+2

#### The fifth number = the summation of two preceding terms

#### 10= 6+4

#### Symbolically,

#### { a }_{ n }={ a }_{ n-1 }+{ a }_{ n-2 } where n>2

#### { a }_{ 5 }={ a }_{ 4 }+{ a }_{ 3 } = 6= 4+2

**Prime Numbers:**

#### However, there are sequences in which no clear pattern or a mathematical formula can be used.

#### Example: 2, 3, 5, 7, 11, 13…

#### In a sequence of prime numbers, no formula can be used to find out the successive terms, but there is a theoretical rule which can be used to define/find the successive terms.

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