Number Sequences
Recognising Number Patterns

This section looks at how the terms of a sequence are related. For example the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, ...

is obtained by adding together two consecutive terms to obtain the next term.

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

Worked Examples

1

The sequence of square numbers is

1, 4, 9, 16, 25, 36, ...

Explain how to obtain the next number in the sequence.

Show me

The next number can be obtained in one of two ways.

(1)

The sequence could be written,

12 , 22 , 32 , 42 , 52 , 62 , ...

and so the next term would be 72 = 49 .

(2)

Calculating the differences between the terms gives

Sequence1,4,9,16,25,36, ...
Difference357911

The difference increases by 2 each time so the next term would be 36 + 13 = 49
2

Describe how to obtain the next term of each sequence below.

(a)

3, 10, 17, 24, 31, ...

Show me

Finding the differences between the terms gives

Sequence3,10,17,24,31,...
Difference7777

All the differences are the same so each term can be obtained by adding 7 to the previous term. The next term would be

31 + 7 = 38
(b)

3, 6, 11, 18, 27, ...

Show me

Here again the differences show a pattern.

Sequence3,6,11,18,27,...
Difference3579

Here the differences increase by 2 each time, so to find further terms add 2 more than the previous difference. The next term would be

27 + 11 = 38
(c)

1, 5, 6, 11, 17, 28, ...

Show me

The differences again help to see the pattern for this sequence.

Sequence1,5,6,11,17,28, ...
Difference415611

If the first difference, 4, is ignored, the remaining pattern of the differences is the same as the sequence itself. This shows that the difference between any two terms is equal to the previous term. So a new term is obtained by adding together the two previous terms. The next term of the sequence will be

17 + 28 = 45
3

A sequence of shapes is shown below.

Write down a sequence for the number of line segments and explain how to find the next number in the sequence.

Show me

The sequence for the number of line segments is

4, 8, 12, 16, ...

The difference between each pair of terms is 4, so to the previous term add 4. Then the next term is 16 + 4 = 20.

This corresponds to the shape opposite, which has 20 line segments.

Exercises

Find the next two terms of each sequence below

(a)
5, 11, 17, 23, 29, ,
(b)
6, 10, 15, 21, 28, ,
(c)
22, 19, 16, 13, 10, ,
(d)
30, 22, 15, 9, 4, ,
(e)
50, 56, 63, 71, 80, ,
(f)
2, 2, 4, 8, 14, 22, ,
(g)
6, 11, 16, 21, 26, ,
(h)
0, 3, 8, 15, 24, ,
(i)
3, 6, 12, 24, 48, ,
(j)
1, 4, 10, 22, 46, ,
(k)
4, −1, −11, −26 , −46, ,

A sequence of numbers is

1, 8, 27, 64, 125, ...

By considering the differences between the terms, find the next two terms.

1, 8, 27, 64, 125, ,

Each sequence of shapes below is made up of lines which join two points.
For each sequence:

(i) write down the number of lines, as a sequence;

(ii) explain how to obtain the next term of the sequence;

(a)
(i)
, , ,
(ii)
(b)
(i)
, , ,
(ii)
(c)
(i)
, , ,
(ii)
(d)
(i)
, , ,
(ii)
(e)
(i)
, , ,
(ii)

Write down a sequence for the number of dots in each pattern. Then find the next number without drawing the next pattern.

(a)

, , , , ,

(b)

, , , , ,

(c)

, , , , ,

(d)

, , , ,

(e)

, , , ,

(f)

, , , , ,

Type ^ sign to use index e.g.: a^b → ab
(a)

Describe how each number in the sequence

1, 4, 9, 16, 25, 36, ...

is formed.

, , , , , , ...

(b)

What is the relationship between the sequence in (a) and the sequences below? For each sequence explain how to calculate the terms.

(i)

3, 6, 11, 18, 27, 38, ...

(i)n =

(ii)

−1, 2, 7, 14, 23, 34, ...

(ii)n =
(iii)

2, 8, 18, 32, 50, 72, ...

(iii)n =
(iv)

4, 16, 36, 64, 100, 144, ...

(iv)n =

Here are the first four numbers of a number pattern.

7, 14, 21, 28, ...

Write down the next two numbers in the pattern.

,

(a)

To generate a sequence of numbers, Paul multiplies the previous number in the sequence by 3, then subtracts 1.

Here are the first four numbers of his sequence.

1, 2, 5, 14, ...

Find the next two numbers in the sequence.

,

(b)

Here are the first four numbers of the sequence of cube numbers.

1, 8, 27, 64, ...

Find the next two numbers in the sequence.

,

The numbers in Row 2 of the above pattern are found by using pairs of numbers from Row 1. For example,

7 = 19 − 12

9 = 28 − 19

(a)

By considering the sequence in Row 2, write down the value of p.

p =
(b)

Find the value of q.

q =