Number Sequences
Formulae for Number Sequences

This section considers how the terms of a sequence can be found using a formula and how a formula can be found for some simple sequences. The terms of a sequence can be described as

u1,  u2,  u3,  u4,  u5, ...

where u1 is the first term, u2 is the second term and so on. Consider the sequence

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

u1 = 1 = 1 × 1
u2 = 4 = 2 × 2
u3 = 9 = 3 × 3
u4 = 16 = 4 × 4
u5 = 25 = 5 × 5

This sequence can be described by the general formula

un = n2

Worked Examples

1

Find the first 5 terms of the sequence defined by the formula un = 3n + 6 .

Show me
u1 = 3 × 1 + 6 = 9
u2 = 3 × 2 + 6 = 12
u3 = 3 × 3 + 6 = 15
u4 = 3 × 4 + 6 = 18
u5 = 3 × 5 + 6 = 21

So the sequence is

9, 12, 15, 18, 21, ...

Note that the terms of the sequence increase by 3 each time and that the formula contains a '3n'.
2

Find the first 5 terms of the sequence defined by the formula

un = 5n − 4

Show me
u1 = 5 × 1 − 4 = 1
u2 = 5 × 2 − 4 = 6
u3 = 5 × 3 − 4 = 11
u4 = 5 × 4 − 4 = 16
u5 = 5 × 5 − 4 = 21

So the sequence is

1, 6, 11, 16, 21, ...

Here the terms increase by 5 each time and the formula contains a '5n'.

In general, if the terms of a sequence increase by a constant amount, d, each time, then the sequence will be defined by the formula

un = dn + c

where c is a constant number.
3

Find a formula to describe each of the sequences below.

(a)

13, 20, 27, 34, 41, 48, ...

Show me

First find the differences between the terms.

Sequence13,20,27,34,41,48, ...
Difference77777

As the difference between each term is always 7, the formula will contain a '7n' and be of the form

un = 7n + c.

To find the value of c consider any term. The first one is usually easiest to use.

Here, using n = 1 and u1 = 13 gives

13 = 7 × 1 + c
13 = 7 + c
c = 6,

so the formula is

un = 7n + 6

Note

Always check that the formula is correct for other terms, e.g. n = 2 and n = 3. In this case,

u2 = 7 × 2 + 6 = 20

and

u3 = 7 × 3 + 6 = 27

so the formula holds.
(b)

1, 12, 23, 34, 45, 56, ...

Show me

Again start by finding the differences between the terms of the sequence.

Sequence1,12,23,34,45,56, ...
Difference1111111111

The difference is always 11 so the formula will contain '11n' and will be

un = 11n + c

Using the first term, i.e. n = 1 and u1 = 1, gives

1 = 11 × 1 + c
1 = 11 + c
c = −10,

so the formula is

un = 11n − 10

[Check: u2 = 11 × 2 − 10 = 12 and u3 = 11 × 3 − 10 = 23, which are correct.]
4

The 7th, 8th and 9th terms of a sequence are 61, 69 and 77 respectively. Find a formula to describe this sequence.

Show me

Looking at the differences,

Sequence61,69,77,...
Difference88

you can conclude that the sequence must be of the form

un = 8n + c

For n = 7, u7 = 56 + c, giving c = 5.

Thus the formula to describe the sequence is

un = 8n + 5

Note

A more general way of tackling this type of problem is to fit the linear sequence

un = dn + c

to the given information.

In this case,

u7 = 7d + c = 61
and u8 = 8d + c = 69

Subtracting u7 from u8 gives d = 8 and substituting for d in either of the two equations gives c = 5.

Thus, as before,

un = 8n + 5

Exercises

Use the formulae below to find the first 6 terms of each sequence.

(a)

un = 4n + 1

, , , , ,
(b)

un = 5n − 7

, , , , ,
(c)

un = 10n + 2

, , , , ,
(d)

un = n2 − 1

, , , , ,
(e)

un = 2n2 + 1

, , , , ,
(f)

un = 2n

, , , , ,

The sequences described by the formulae,

A   un = 8n − 2,
B   un = 3n + 5,
C   un = n2 + 1,
D   un = n3 − 1

are given below. Select the formula that describes each sequence.

(a)

0, 7, 26, 63, 124, ...

(b)

8, 11, 14, 17, 20, ...

(c)

6, 14, 22, 30, 38, ...

(d)

2, 5, 10, 17, 26, ...

Find the 20th term for each sequence below.

(a)
un = 4n u20 =
(b)
un = 3n − 50 u20 =
(c)
un = 4n + 7 u20 =
(d)
un = 2n − 40 u20 =
(e)
un = n2 − 4 u20 =
(f)
un = u20 =

By considering the sequence described by un = n3, find formulae to describe the following sequences.

(a)
0, 7, 26, 63, 124, ... un =
(b)
10, 17, 36, 73, 134, ... un =
(c)
199, 192, 173, 136, 75, ... un =

The 10th, 11th and 12th terms of a sequence are 50, 54 and 58.

Find a formula to describe this sequence and write down the first 5 terms.

un =

, , , ,

The 100th, 101st and 102nd terms of a sequence are 608, 614 and 620.

Find a formula to describe this sequence and find the 10th term.

un =

u10 =

Here are the first four terms of a number sequence.

7, 11, 15, 19

Write down the nth term of the sequence.

un =

Patterns of squares are formed using sticks. The first three patterns are drawn below.

The table shows the number of sticks needed for each pattern.

(a)

How many sticks are needed for Pattern 4?

sticks
(b)

How many more sticks are needed to make Pattern 5 from Pattern 4?

more sticks
(c)

There is a rule for finding the number of sticks needed to make any of these patterns of squares.
If the number of squares in a pattern is s, write down the rule.

number =

Challenge!

John and Julie had a date one Saturday. They agreed to meet outside the cinema at 8 pm.

Julie thought that her watch was 5 minutes fast but in actual fact it was 5 minutes slow.

John thought that his watch was 5 minutes slow but in actual fact it was 5 minutes fast.

Julie deliberately turned up 10 minutes late while John decided to turn up 10 minutes early.

Who turned up first and how long had he/she to wait for the other to arrive?