Linear sequences

Gillian has a pay-as-you-go mobile phone contract and she is trying to work out how much money she has spent on texts this week.

The cost of 1 text message is 5p.

You would make a table of values for up to 6 texts showing the cost in pence as below.

Table showing cost of texts increasing by 5

Now, we need to find a formula that will help us find the cost of any number of texts.

From the table above, you should notice that the cost increases by 5 each time.

The formula is \(C=5\times T\)


Gillian has sent 87 texts this week. How much money has Gillian spent on texts this week?

Cost (in pence) = 5 x number of texts

This can be re-written as:

\[C = 5 \times T\]

Gillian sent 87 texts this week. We can now use this formula to calculate the cost.

\[C = 5 \times T\]

\[C = 5 \times 87\]

\[C = 435p = \pounds4.35\]

The nth term

1, 4, 7, 10 is a sequence starting with 1.

You get the next term by adding 3 to the previous term.

You are often asked to find a formula for the nth term.

Sequence formula example increasing by 3 each time


Find the nth term.

Sequence formula example increasing by 3 each time

To do this, we first of all find the difference between each term. This tells us part of our formula:

\[S = 3 \times n\]

When we substitute \(n = 1\) into this formula, we find that it doesn’t work as \(S = 3 \times 1 = 3\), but the first term is 1.

Now all we have to do is subtract 2.

\[S = 3 \times 1 - 2\]

\[S = 1\]

Now try this for some other terms to make sure your rule works:

Term 2

\[S = 3 \times 2 - 2\]

\[S = 6 - 2\]

\[S = 4\,Correct!\]

Term 4

\[S = 3x4 -2\]



Now we have established our formula:

\[S = 3 \times n - 2\]


\[S = 3n - 2\]

This method will always work for sequences where the difference between terms stays the same.


Find the nth term in the sequence 1, 5, 9, 13.

nth term12345
  • First, find the difference between each term
  • The difference between each term is 4
  • This lets you work out the first part of the formula
  • The formula for this sequence will start with \(S = 4n\)
  • Now look at each term
  • When n = 1, \(S = 4 \times 1 = 4\). But the 1st term is 1.
  • We therefore have to subtract 3 to get the first correct term and also check that this works for the 2nd term.

The formula for the sequence is \(S = 4n - 3\).