Number sequences


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.


Counting or whole numbers


This is easy!

1

2

3

4

5

6

7

8

9

...


Odd and even numbers


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.


Square numbers

1

4

9

16

25

36

49

64

81

...


This sequence is derived from the following calculations


  • 1² = 1 x 1 = 1
  • 2² = 2 x 2 = 4
  • 3² = 3 x 3 = 9
  • 4² = 4 x 4 = 16
  • 5² = 5 x 5 = 25
  • 6² = 6 x 6 = 36
  • 7² = 7 x 7 = 49
  • 8² = 8 x 8 = 64
  • 9² = 9 x 9 = 81 ...


Note


nth term (square numbers) =  n²

e.g. 6th term  =  6²  =  6 x 6  =  36


For more on square numbers click here.


Cube numbers

1

8

27

64

125

216

343

512

729

...


Again, here are the calculations


  • 1³ = 1 x 1 x 1 = 1
  • 2³ = 2 x 2 x 2 = 8
  • 3³ = 3 x 3 x 3 = 27
  • 4³ = 4 x 4 x 4 = 64
  • 5³ = 5 x 5 x 5 = 125
  • 6³ = 6 x 6 x 6 = 216
  • = 7 x 7 x 7 = 343
  • 8³ = 8 x 8 x 8 = 512
  • = 9 x 9 x 9 = 729 ...


Note


nth term (cube numbers) =  n³

e.g. 3rd term  =  3³  =  3 x 3 x 3  =  27


For more on cube numbers click here.


Triangular numbers

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


Increasing difference


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.



Decreasing difference


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


Fibonacci sequence

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

  • 0 + 1 = 1
  • 1 + 1 = 2
  • 1 + 2 = 3
  • 2 + 3 = 5
  • 3 + 5 = 8
  • 5 + 8 = 13
  • 8 + 13 = 21 ...


Summary of number sequences


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:

Arithmetic sequences

Geometric sequences


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