Sequences
A sequence is a list of numbers in a particular order. In GCSE Maths, you need to continue sequences, describe rules and find nth term formulae.
A sequence might look like:
4, 7, 10, 13, 16, ...
This sequence goes up by 3 each time.
Term-to-term rules
A term-to-term rule tells you how to get from one term to the next.
Example 1: Continue a sequence
Continue the sequence: 5, 9, 13, 17, ...
The sequence increases by 4 each time.
17 + 4 = 21
The next term is 21.
Example 2: Find the term-to-term rule
Find the rule for 20, 17, 14, 11, ...
Each term decreases by 3.
The term-to-term rule is subtract 3.
Position-to-term rules
A position-to-term rule tells you how to find a term using its position in the sequence. The nth term is a formula for any term in the sequence.
If the nth term is 3n + 2:
1st term = 3(1) + 2 = 5
2nd term = 3(2) + 2 = 8
3rd term = 3(3) + 2 = 11
Finding the nth term of a linear sequence
For a linear sequence, the difference between terms is constant. This difference becomes the number in front of n.
Example 3: Find the nth term
Find the nth term of 4, 7, 10, 13, ...
The sequence increases by 3, so start with:
3n
The sequence 3n gives 3, 6, 9, 12, ...
To get 4, 7, 10, 13, add 1.
3n + 1
The nth term is 3n + 1.
Example 4: Another nth term
Find the nth term of 8, 13, 18, 23, ...
The difference is 5, so start with:
5n
5n gives 5, 10, 15, 20, ...
To get the original sequence, add 3.
5n + 3
Using the nth term
Once you have the nth term, you can find any term by substituting the term number into the formula.
Example 5: Find a term from the nth term
The nth term is 4n - 1. Find the 10th term.
Substitute n = 10:
4(10) - 1 = 40 - 1 = 39
The 10th term is 39.
Checking if a number is in a sequence
To check whether a number appears in a sequence, set the nth term equal to that number and solve.
Example 6: Is 47 in the sequence 5n + 2?
Set the nth term equal to 47:
5n + 2 = 47
Subtract 2:
5n = 45
Divide by 5:
n = 9
Since n is a positive whole number, 47 is in the sequence.
A common mistake is using the common difference as the full nth term. If the difference is 3, the nth term starts with 3n, but you may still need to add or subtract a number.
To check your nth term, substitute n = 1, 2 and 3. If it gives the first three terms correctly, your rule is likely to be right.
Video explanation
A short Worthing Maths Tutor video explanation for sequences can be embedded here later to improve student engagement and time on page.
Practice questions
- Continue the sequence: 6, 10, 14, 18, ...
- Find the term-to-term rule for 30, 25, 20, 15, ...
- Find the nth term of 2, 5, 8, 11, ...
- Find the nth term of 7, 12, 17, 22, ...
- The nth term is 6n - 4. Find the 12th term.
Answers
- 22
- Subtract 5
- 3n - 1
- 5n + 2
- 68
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