Sequences

A sequence is a set of numbers arranged in a specific order.

A sequence has some key features:

  • Each number in a sequence is called a term of the sequence
  • Each term has a position, eg 1st, 2nd, 3rd etc
  • The terms are linked by a pattern or rule
  • The three dots at the end, ..., show that the sequence can keep going and more terms worked out

Here are some well-known number sequences that you should learn to recognise:

Even numbers

The terms are 2, 4, 6, 8...

Their positions are 1st, 2nd, 3rd, 4th...

To find the next term the rule is +2

Question

What are the 5th and 6th terms?

5th term = 4th term + 2

= 8 + 2

= 10

6th term = 5th term + 2

= 10 + 2

= 12

Odd numbers

The terms are 1, 3, 5, 7...

The positions are 1st, 2nd, 3rd, 4th...

To find the next term the rule is +2

Question

What are the 5th and 6th terms?

5th term = 4th term + 2

= 7 + 2

= 9

6th term = 5th term + 2

= 9 + 2

= 11

Square numbers

The terms are 1, 4, 9, 16...

The positions are 1st, 2nd, 3rd, 4th...

The rule is each term is formed by multiplying the position by itself:

Square numbers diagram
Question

What are the 5th and 6th terms?

5th term = 5 x 5

= 25

6th term = 6 x 6

= 36

Cube numbers

The terms are 1, 8, 27, 64...

The positions are 1st, 2nd, 3rd, 4th...

The rule is each term is formed by multiplying the position by itself three times:

Cube numbers diagram
Question

What are the 5th and 6th terms?

5th term = 5 x 5 x 5

= 125

6th term = 6 x 6 x 6

= 216

Triangle numbers

The terms are 1, 3, 6, 10...

The positions are 1st, 2nd, 3rd, 4th...

The rule is that each term is previous term + position, eg:

4th term = 3rd term + 4

Red dots that are grouped in series of 1, 3, 6, 10, 15, 21

Question

What are the 5th and 6th terms?

5th term = 4th term + 5

= 10 + 5

= 15

6th term = 5th term + 6

= 15 + 6

= 21

Powers of 2

The terms are 2, 4, 8, 16...

(2, 2 x 2, 2 x 2 x 2, 2 x 2 x 2 x 2...)

The positions are 1st, 2nd, 3rd, 4th...

To find the next term, the rule is x 2

Question

What are the 5th and 6th terms?

5th term = 4th term x 2

= 16 x 2

= 32

6th term = 5th term x 2

= 32 x 2

= 64

Powers of 10

The terms are 10, 100, 1000, 10000...

(10, 10 x 10, 10 x 10 x 10, 10 x 10 x 10 x 10...)

The positions are 1st, 2nd, 3rd, 4th...

To find the next term, the rule is x 10

Question

What are the 5th and 6th terms?

5th term = 4th term x 10

= 10,000 x 10

= 100,000

6th term = 5th term x 10

= 100,000 x 10

= 1,000,000