# 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

## 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

## 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 = 11 + 1 = 21 + 2 = 32 + 3 = 53 + 5 = 85 + 8 = 138 + 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: