# 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.

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 $$C=5\times T$$

Question

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.

Question

Find the nth term.

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

$S = 3x4 -2$

$S=12-2$

$S=10\,Correct!$

Now we have established our formula:

$S = 3 \times n - 2$

or:

$S = 3n - 2$

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

Question

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

 nth term Sequence 1 2 3 4 5 1 5 9 13 17
• 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 $$S = 4n - 3$$.