Number sequences are sets of numbers which follow specific patterns  hence their alternate name number patterns.
Each number in a sequence or pattern is called a term.
Let's consider some examples to help you recognize different sequences or patterns.
This is easy!
1 
2 
3 
4 
5 
6 
7 
8 
9 
... 
Here's the start of both of these sets of numbers
1 
3 
5 
7 
9 
11 
13 
15 
17 
... 
2 
4 
6 
8 
10 
12 
14 
16 
18 
... 
Note
nth term (odd numbers) = 2n  1
e.g. 5th term = (2 x 5)  1 = 10  1 = 9
nth term (even numbers) = 2n
e.g. 7th term = 2 x 7 = 14
Each of the above are examples of arithmetic sequences  there's a common difference between successive terms, +1 for counting numbers, +2 for odd and even numbers.
1 
4 
9 
16 
25 
36 
49 
64 
81 
... 
This sequence is derived from the following calculations
Note
nth term (square numbers) = n²
e.g. 6th term = 6² = 6 x 6 = 36
For more on square numbers click here.
1 
8 
27 
64 
125 
216 
343 
512 
729 
... 
Again, here are the calculations
Note
nth term (cube numbers) = n³
e.g. 3rd term = 3³ = 3 x 3 x 3 = 27
For more on cube numbers click here.
1 
3 
6 
10 
15 
21 
28 
36 
45 
... 
Perhaps the easiest way to understand this sequence is to imagine arranging bowling pins into a triangular pattern.
Here's a representation of the first four terms in the sequence
Note
nth term (triangular numbers) = n(n + 1)/2
e.g. calculate the 4th term
n = 4
n(n + 1) = 4(4 + 1) = 4 x 5 = 20
4th term = 20/2 = 10
Triangular numbers are an example of a sequence in which there's an increasing difference between successive terms
1 
3 
6 
10 
15 
Look closely and you'll see the difference between successive terms increases by one each time you take a step along the sequence
1 
+2 
3 
+3 
6 
+4 
10 
+5 
15 
Now check back and make sure you're happy this increasing difference holds true for further terms along the triangular number sequence.
There may also be a decreasing difference between successive terms in a number sequence.
For instance, consider the following pattern
15 
10 
6 
3 
1 
The difference between successive terms decreases by one each time you take a step along the sequence
15 
5 
10 
4 
6 
3 
3 
2 
1 
0 
1 
1 
2 
3 
5 
8 
13 
21 
... 
In this particular sequence, the first two terms are 0 and 1, and each subsequent term is found by adding together the previous two terms

As you can see, there are several forms of number sequence, but by inspecting the terms of a sequence and the differences between them you can usually work out the sequence pattern.
For more on number sequences click: