Sequences 1 (Edexcel GCSE Maths): Revision Notes
Sequences 1
What is a sequence?
A sequence is a pattern of numbers that follows a specific rule. Each individual number in a sequence is called a term.
Understanding sequences helps you spot patterns and predict what comes next. This is a fundamental skill in algebra that appears frequently in GCSE exams.
Sequences are everywhere in mathematics and real life - from the number of petals on flowers to the arrangement of seeds in sunflowers. Mastering sequences will give you a strong foundation for more advanced mathematical concepts.
Common types of sequences
Even number sequences: 2, 4, 6, 8, 10...
- Each term increases by 2
Square number sequences: 1, 4, 9, 16, 25...
- Each term is a perfect square (...)
Arithmetic sequences: Numbers that increase or decrease by the same amount each time
- Example: 11, 15, 19, 23, 27, 31...
- The difference between consecutive terms is +4
Finding missing terms in sequences
To find missing terms, you need to identify the rule that connects one term to the next.
Method:
- Look at the differences between consecutive terms
- Check if the difference is constant
- Apply the same rule to find missing terms
Worked Example: Finding Missing Terms
In the sequence 11, 15, 19, 23, 27, 31...
Step 1: Find the differences
Step 2: Identify the rule The rule is "add 4 to the previous term"
Step 3: Apply to find missing terms If we need the 7th term:
Working backwards through sequences
Sometimes you're given a term in the middle of a sequence and need to find earlier terms. You can use function machines to work backwards.
Using function machines:
- If the forwards rule is "+6 then ÷2", the backwards rule is "×2 then -6"
- Always reverse both the operations and their order
Worked Example: Working Backwards
If the 3rd term is 8 and the rule is "add 6 to the previous term then divide by 2":
Step 1: Start with the 3rd term = 8
Step 2: Work backwards to find 2nd term
Step 3: Work backwards to find 1st term
Step 4: Check by working forwards
- 1st term:
- 2nd term: ✓
- 3rd term: ✓
Generating sequences using the nth term
You can find any term in a sequence by using the nth term formula. This tells you the value of the nth term without having to work out all the previous terms.
Common nth term patterns:
| nth term formula | Example terms |
|---|---|
| 7, 5, 3, 1... | |
| 11, 14, 19, 26... | |
| 49, 46, 41, 34... |
How to use nth term formulas:
- Substitute the term number (n) into the formula
- Calculate the result
- This gives you the value of that term
Worked Example: Using nth Term Formula
For the formula :
Finding the 7th term:
Finding the 8th term:
The 8th term is the first negative term in this sequence, showing how the sequence changes from positive to negative values.
Special sequences: Fibonacci-type
A Fibonacci-type sequence is where each term is found by adding the two previous terms together.
Example: 2, 3, 5, 8, 13...
- Rule: "Add two consecutive terms to get the next term"
- , ,
These sequences appear frequently in nature and are important in mathematics. The famous Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) follows this same pattern and can be found in flower petals, pinecones, and spiral shells.
Common Mistakes to Avoid:
- Don't forget to show your working when finding missing terms
- Always check your answers by working forwards or backwards
- Be careful with negative numbers when using nth term formulas
- Read the question twice to ensure you're finding the right term
Key Points to Remember:
- A sequence is a pattern of numbers following a rule, and each number is called a term
- Find missing terms by identifying the pattern or rule between consecutive terms
- Use function machines to work backwards through sequences by reversing operations and their order
- nth term formulas let you find any term directly without calculating all previous terms
- Fibonacci-type sequences are formed by adding the two previous terms together