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 | 1 | 2 | 3 | 4 | 5 |
| Sequence | 1 | 5 | 9 | 13 | 17 |
| nth term |
|---|
| 1 |
| 2 |
| 3 |
| 4 |
| 5 |
| Sequence |
|---|
| 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 \(S = 4n - 3\).
