Sequences 2 (AQA GCSE Maths): Revision Notes
Sequences 2
What are arithmetic sequences?
An arithmetic sequence (also called a linear sequence) is a sequence of numbers where the difference between consecutive terms remains constant. This constant difference is what makes the sequence predictable and allows us to create a formula.
In your GCSE exam, you'll need to work out the nth term of arithmetic sequences and use this formula to find specific terms or check whether certain numbers belong to the sequence.
Understanding arithmetic sequences is fundamental for GCSE success. You'll encounter these in various question types, from simple term finding to complex problem-solving scenarios.
Finding the nth term formula
To find the formula for the nth term, you need to identify the common difference between consecutive terms.
Worked Example: Finding the nth term
Consider this sequence: 1, 5, 9, 13, 17...
Step 1: Calculate the common difference
- Between each pair of consecutive terms, the difference is +4
- This tells us the sequence increases by 4 each time
Step 2: Write out the pattern
- 1st term: ✓
- 2nd term: ✓
- 3rd term: ✓
Step 3: Create the general formula
- The nth term =
Step 4: Check your formula Always verify by substituting values of n back into your formula to ensure you get the correct sequence terms.
Using the nth term formula
Once you have the formula, you can generate any term in the sequence.
Quick Example: Using nth term = :
- 20th term =
This demonstrates how powerful the nth term formula is for finding any position in the sequence efficiently.
Checking if a number belongs to a sequence
A crucial exam skill is determining whether a specific number is part of an arithmetic sequence.
Remember: n must always be a positive whole number. If your calculation gives you a negative number, zero, or a decimal for n, then the number is not part of the sequence.
Method 1: Trial and error approach
Test consecutive integer values of n until you find the target number or determine it's impossible.
For checking if 99 is in the sequence with nth term = :
- When :
- When :
- Since 99 falls between 97 and 101, and we can only use whole number values of n, 99 is not a term in this sequence.
Method 2: Algebraic approach
Set up an equation and solve for n.
For the same example:
- Set
- Solve: , so
- Since 25.5 is not a whole number, 99 is not a valid term in the sequence.
Key Points to Remember:
- Arithmetic sequences have a constant difference between consecutive terms
- Find the common difference first, then work out the general pattern
- The nth term formula lets you find any term in the sequence
- Always check your formula by substituting small values of n
- For a number to be in the sequence, n must be a positive whole number
- Double-check your nth term formula by substituting , , etc.
- When writing your final answer, state clearly whether the number is or isn't in the sequence and explain why