Sequences Revision Notes for AQA GCSE Further Maths

Sequences

What is a sequence?

A sequence is a list of numbers that follow a particular pattern or rule. Each number in the sequence is called a term. The position of each term is identified by a number called n, where the first term has position n = 1, the second term has position n = 2, and so on.

For example, here are the first few terms of a sequence:

4, 10, 16, 22, 28, ...

The first term is 4, each subsequent term can be obtained by adding 6 to the previous term.

infoNote

The nth term of a sequence is an expression that tells us the rule for finding any term in the sequence. When we know the nth term formula, we can substitute any value for n to find the corresponding term.

Finding terms from the nth term formula

When you're given the nth term formula, you can find any specific term by substituting the position number into the formula.

lightbulbExample

Worked Example: Finding Terms from Formula

If the nth term = $n^2 + \frac{30}{n}$ , find the first three terms and the 10th term.

Solution:

1st term = $1^2 + \frac{30}{1} = 1 + 30 = 31$
2nd term = $2^2 + \frac{30}{2} = 4 + 15 = 19$
3rd term = $3^2 + \frac{30}{3} = 9 + 10 = 19$
10th term = $10^2 + \frac{30}{10} = 100 + 3 = 103$

Linear sequences

A linear sequence (also called an arithmetic sequence) is one where there is a constant difference between consecutive terms. This constant difference is what makes the sequence "linear" - if you were to plot the terms on a graph, they would form a straight line.

chatImportant

The nth term of a linear sequence has the form: nth term = an + b

where a and b are constants:

a is the common difference between consecutive terms
b is a constant that helps determine the starting value

Finding the nth term of a linear sequence

There are two main methods for finding the nth term formula of a linear sequence:

Method 1: Using two equations

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Worked Example: Method 1 - Two Equations

Find the nth term of the sequence: 3, 12, 21, 30, 39, ...

Step 1: Set up the general form nth term = an + b

Step 2: Use two known terms to create equations

When n = 1, the term is 3: $3 = a(1) + b → 3 = a + b$ ... ①
When n = 2, the term is 12: $12 = a(2) + b → 12 = 2a + b$ ... ②

Step 3: Solve the simultaneous equations

Subtract equation ① from equation ②: $9 = a$
Substitute back: $3 = 9 + b$ , so $b = -6$

Therefore: nth term = $9n - 6$

Method 2: Using the common difference

lightbulbExample

Worked Example: Method 2 - Common Difference

Using the same sequence: 3, 12, 21, 30, 39, ...

Step 1: Find the common difference between consecutive terms

Common difference = $12 - 3 = 9$
This becomes the coefficient of n, so $a = 9$

Step 2: Use the first term to find b

When n = 1, the term is 3: $3 = 9(1) + b$
So $3 = 9 + b$ , which gives $b = -6$

Therefore: nth term = $9n - 6$

Working with decreasing sequences

lightbulbExample

Worked Example: Decreasing Sequence

Find the nth term of the sequence: 15, 13, 11, 9, ...

The common difference is $13 - 15 = -2$ (negative because the sequence is decreasing)

Using Method 2:

The coefficient of n is -2
When n = 1: $15 = -2(1) + b$ , so $b = 17$
Therefore: nth term = $-2n + 17$ (which can also be written as $17 - 2n$ )

chatImportant

For a linear sequence that decreases, the common difference is negative.

Quadratic sequences

A quadratic sequence is one where the nth term contains $n^2$ . The general form is:

$\text{nth term} = an^2 + bn + c$

where a, b, and c are constants.

infoNote

The key property of quadratic sequences is that while the first differences (between consecutive terms) vary, the second differences are always constant.

Finding the nth term of a quadratic sequence

The method involves working out the differences between consecutive terms:

lightbulbExample

Worked Example: Finding Quadratic nth Term

Find the nth term of the quadratic sequence: 3, 6, 13, 24, 39, ...

Step 1: Work out the second differences

First differences: 3, 7, 11, 15
Second differences: 4, 4, 4 (constant)

Step 2: Find the coefficient of $n^2$

The coefficient of $n^2$ is always half of the second difference
Coefficient of $n^2 = 4 ÷ 2 = 2$

Step 3: Compare with $2n^2$ sequence

Original sequence: 3, 6, 13, 24, 39
$2n^2$ sequence: 2, 8, 18, 32, 50
Differences needed: 1, -2, -5, -8, -11

Step 4: Find the linear part

The sequence of differences (1, -2, -5, -8, -11) is linear with nth term = $-3n + 4$

Step 5: Combine the parts

nth term = $2n^2 + (-3n + 4) = 2n^2 - 3n + 4$

chatImportant

Key insight for quadratic sequences:

For any quadratic sequence, all the second differences will be the same. This constant second difference is always twice the coefficient of $n^2$ in the nth term formula.

bookmarkSummary

Key Points to Remember:

Linear sequences have a constant difference between consecutive terms, and their nth term has the form $an + b$
Method 1 for linear sequences uses two equations with known terms to find a and b
Method 2 for linear sequences uses the common difference as the coefficient of n
Quadratic sequences have constant second differences, and their nth term has the form $an^2 + bn + c$
The coefficient of $n^2$ in a quadratic sequence is always half the constant second difference
When finding quadratic nth terms, compare your sequence with the pure $n^2$ sequence to identify the linear adjustment needed

Sequences (AQA GCSE Further Maths): Revision Notes

Sequences

What is a sequence?

Finding terms from the nth term formula

Linear sequences

Finding the nth term of a linear sequence

Working with decreasing sequences

Quadratic sequences

Finding the nth term of a quadratic sequence

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