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

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:

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=2×1=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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