- Number sequences are the sequences of numbers in which the successive numbers are linked with specific rule.
- If we found that rule then we can find the next numbers in that sequence.
- It is very interesting to learn about sequences in mathematics and to identify the number sequences.
- Sometimes in given sequence there will be a number is missing and we have to find that number by finding the rule linked with them.

**Number sequences are classified as:**

- Arithmetic sequence
- Geometric sequence
- Arithmetic- Geometric sequence
- Geometric- Arithmetic sequence
- Prime sequence
- Power sequence
- Reversal sequence
- Two-tier sequence
- Twin sequence
- Fibonacci sequence

**1.) Arithmetic sequence:**

- In arithmetic sequence, the successive numbers are obtained only by adding a particular number or fixed number in the previous number.
- In arithmetic sequence, the fixed number by adding which we get the next number is called as the common difference d.

**For example:**

- Consider the sequence 3, 7, 11, 15, …

Here, the common difference d= 4

Since, 3+4= 7, 7+4=11, 11+4=15

In this way by adding common difference d=4 here we can obtain the next numbers.

The next number after 15 here will be 15+4= 19, 19+4= 23 and so on.

Hence, this is the arithmetic sequence.

- Now, consider the sequence 21, 17, 13, 9, 5,…

Here, the common difference d= -4

Since, 17-21=-4, 13-17=-4, 9-13=-4 and so on.

Thus, by adding common difference d=-4 in the next numbers we can get the sequence.

21-4=17, 17-4=13, 13-4=9, 9-4=5, 5-4=1 and so on.

Hence, this is the arithmetic sequence.

- Consider the sequence 3.6, 4.1, 4.6, 5.1,…and so on.

Here the common difference d= 0.5

Since, 3.6+0.5= 4.1, 4.1+0.5= 4.6, 4.6+0.5= 5.1 and 5.1+0.5= 5.6

In this way by adding common difference d=0.5 we can find the next numbers in the sequence.

Hence, this is also the arithmetic sequence.

**2.) Geometric sequence:**

- In geometric sequence, by multiplying with a specific number or fixed number to a previous number we can get the next numbers.
- And the fixed number by multiplying it with the previous number we get the next number is called as common ratio.

**For example:**

- Consider a simple sequence 3, 6, 12, 24,…and so on

Here, the common ratio is r= 2

Since, 3*2= 6, 6*2=12, 12*2=24 and 24*2=48 and so on.

In this way by multiplying with 2 to each number we get the next Numbers.

Hence, this is the geometric sequence.

- Consider the sequence 3, -9, 27, -81,…and so on.

Here also the common ratio is r=-3.

Since, 3*(-3)=-9, -9*(-3)=27, 27*(-3)=-81 and so on.

In this way, we can find the next numbers only by multiplying with -3 to the previous number.

Hence, this is the geometric sequence.

- Consider the sequence 400, 200, 100, 50, 25,…and so on.

Here, the common ratio r=1/2.

Since, 400/200= 1/2, 200/100=1/2, 100/50=1/2, 50/25=1/2.

So next number can be obtained by 25*1/2= 12.5 and so on.

Hence this is the geometric sequence.

**3.) Arithmetic- geometric sequence:**

- In arithmetic geometric sequence, each number is in the form (x + a)*b that means we have to first add a fixed number ‘a’ and then we have to multiply it by a fixed number ‘b’ and then we get the next numbers of the sequence.
- This type of sequence is called as the arithmetic- geometric sequence.

**For example:**

- 2, 12, 42, 132, 402,…and so on

Here, a= 2 and b= 3

Since, (2+2)*3= 4*3=12

(12+2)*3=14*3=42

(42+2)*3=44*3=132

(132+2)*3= 402…and so on.

In this way, first adding a fixed number and then multiplying with again a fixed number we get the next numbers.

Hence, this sequence is called as arithmetic geometric sequence.

**4.) Geometric- arithmetic sequence:**

- In geometric- arithmetic sequence, each number is in the form (x*a)+b that means we have to first multiply by a fixed number ‘a’ and then we have to add a fixed number ‘b’ and then we get the next numbers.
- This type of sequence is called as geometric- arithmetic sequence.

**For example:**

- Consider the sequence 3, 7, 15, 31, 63,…and so on.

Here, a=2 and b=1

(3*2)+1=6+1=7

(7*2)+1= 14+1=15

(15*2)+1=30+1=31

(31*2)+1=62+1=63 and so on.

In this way, by multiplying with fixed number a=2 and after that adding a fixed number b=1 we get the next numbers and hence this type sequence is called as the geometric- arithmetic sequence.

**5.) Prime sequence:**

- Prime sequence is the sequence which involves only prime numbers and that successive prime numbers are linked with some specific rule.
- Prime numbers are those numbers which are divisible by 1 and the number itself.

**For example:**

- 7, 11, 13, 17, 19,…and so on.

This is a prime sequences which includes all the successive prime numbers starting from 7.

The next prime number after 19 here is 23.

**6.) Power sequence:**

- In power sequence, every term is the nth power of some consecutive numbers.
- The power sequence may be square sequence, cube sequence or may be higher order sequence.

**For example:**

- 4, 9, 16, 25, 36,…and so on.

This is square sequence because the numbers in this sequence are the squares of consecutive numbers 2, 3, 4, 5, 6,…and so on.

Hence, the next number in this sequence will be square of 7= 49 then next number is the square of 8= 64 and so on.

Thus, the resultant power sequence will be 4, 9, 16, 25, 36, 49, 64, 81,…and so on.

This type of sequence is called as power sequence.

- Consider the cube sequence 27, 64, 125, 216, 343,…and so on.

In this sequence the consecutive numbers are the cube of the numbers 3, 4, 5, 6, 7,…and so on.

Hence, this is a cube sequence called as power sequence.

**7.) Reversal sequence:**

- In this sequence, if we reversed the numbers of this sequence then we may get any sequence arithmetic, geometric, arithmetic-geometric, geometric arithmetic, prime or power sequence.
- Such sequence is called as reversal sequence.

**For example:**

- Consider the sequence 71, 02, 32, 62, 92,…and so on.

If we reversed the numbers in this sequence then we get the another sequence as 17, 20, 23, 26, 29,…and so on.

Here this is the arithmetic sequence with common difference d=3.

Thus, the next number will be here 29+3= 32 and then 32+3=35 and so on.

Thus we get, 17, 20, 23, 26, 29, 32, 35,…and so on.

By reversing the numbers in it we get our real reversal sequence as 71, 02, 32, 62, 92, 23, 53,…and so on.

This type of sequence is called as reversal sequence.

- Consider the another sequence 61, 52, 63, 94, 46, 18,…and so on.

If the numbers in a sequence are randomised and not consecutive then that sequence must be the reversal sequence.

Here, by reversing the numbers of the sequence we get it as 16, 25, 36, 49, 64, 81,…and so on which is the square sequence called as power sequence.

Hence, the next term will be here 100, 121,…and so on.

Hence our sequence will be 16, 25, 36, 49, 64, 81, 100, 121,…and so on.

Hence, our resultant reversal sequence will be 61, 52, 63, 94, 46, 18, 001, 121,…and so on.

Such type of sequence is called as the reversal sequence.

**8.) Two – tier sequence:**

In two tier sequence, the difference between the consecutive numbers forms again the sequence from which we can find the next numbers of that sequence such sequence is called as the two – tier sequence.

**For example:**

- Consider the sequence 1, 4, 10, 19, 31,…and so on.

If we have find the difference between the consecutive numbers in this sequence then we get as (4-1), (10-4), (19-10), (31-19),…which is nothing but the 3, 6, 9, 12,…and so on.

But this is the arithmetic sequence with common difference d= 3.

Hence, next numbers will be 12+3= 15 and 15+3=18 and so on.

So we get as 3, 6, 9, 12, 15, 18,…and so on.

Hence, our resultant two – tier sequence will be 1, 4, 10, 19, 31, (31 + 15),…and so on.

That means 1, 4, 10, 19, 31, 46,…and so on.

This is the two – tier sequence.

**9.) Twin sequence:**

In twin sequence as the name there are two sequences are mixed in one another.

**For example:**

- Consider the sequence 1, 0, 6, 3, 11, 6, 16,…and so on.

In this sequence there are two sequences are packed.

One is the 1, 6, 11, 16,…and so on which is the arithmetic sequence with common difference d=5 and hence the next number will be 16+5=21.

And the second sequence is 0, 3, 6,…and so on which is also the arithmetic sequence with common difference d=3 and hence next number will be 6+3=9.

By combining both the sequences such that the numbers are alternately occurring such sequence is called as twin sequence.

Hence, our resultant twin sequence will be 1, 0, 6, 3, 6, 16, 9, 21,…and so on.

**10.) Fibonacci sequence:**

Fibonacci sequence is the naturally occurring sequence having first two terms as 1, 1 and each number after that is obtained by summing the previous two numbers.

For example:

- 1, 1, 2, 3, 5, 8, 13, 21,…and so on.

If we observe here, then we see that each number is obtained by adding previous two numbers.

Hence next number will be 13+21= 34

In this we get a sequence which is called as the Fibonacci sequence.

The somewhat reason behind nature’s beauty is Fibonacci sequence.