Sequences (Edexcel GCSE Maths): Revision notes

Section 1

How do you generate sequences from rules?

Sequences can be generated using either term-to-term rules (recurrence relations) or position-to-term rules (explicit formulas).

Term-to-term rules describe how to find each term from the previous term(s). For example, if the rule is "add 3 each time", starting from 2: the sequence is 2, 5, 8, 11, 14, ...

Position-to-term rules give a formula for the nth term. If the rule is $a_n = 2n + 1$ , then:

When $n = 1$ : $a_1 = 3$
When $n = 2$ : $a_2 = 5$
When $n = 3$ : $a_3 = 7$

Both methods should produce the same sequence; choose whichever rule you are given and apply it systematically.

Key termsterm-to-term ruleposition-to-term rulerecurrence relation

Example

Generate the first four terms of the sequence with term-to-term rule "multiply by 2, then subtract 1" starting from 5. Answer: $a_1 = 5$ , $a_2 = 2(5) - 1 = 9$ , $a_3 = 2(9) - 1 = 17$ , $a_4 = 2(17) - 1 = 33$ . Sequence: 5, 9, 17, 33.

Section 2

What is the nth term formula for linear sequences?

A linear sequence has a constant difference between consecutive terms. The nth term formula is:

$a_n = an + b$

where $a$ is the common difference and $b$ is a constant.

To find the nth term formula:

Calculate the difference between consecutive terms (this is $a$ ).
Use the formula $a_n = an + b$ .
Substitute one known term to find $b$ . For example, if $a_1 = 3$ and $a = 2$ , then $3 = 2(1) + b$ , so $b = 1$ .
Write the formula: $a_n = 2n + 1$ .

Example sequence: 5, 8, 11, 14, ...

Common difference: $8 - 5 = 3$
Using $a_1 = 5$ : $5 = 3(1) + b$ , so $b = 2$
Formula: $a_n = 3n + 2$

Key termslinear sequencenth term formulacommon difference

Exam tip

Examiners expect you to show the calculation of the common difference and clearly state the formula $a_n = an + b$ with both constants identified numerically. Always verify your formula by checking it against at least two terms from the sequence.

Common mistake

Students often forget to find $b$ correctly. Remember: substitute the position and term value into $a_n = an + b$ , not just write down $b$ as the first term.

Section 3

What are geometric sequences and other special sequences?

A geometric sequence has a constant common ratio between consecutive terms. Each term is found by multiplying the previous term by a fixed number $r$ .

Linear sequence: constant difference (e.g. 2, 5, 8, 11, ... with difference 3).

Geometric sequence: constant ratio (e.g. 3, 6, 12, 24, ... with ratio 2).

Fibonacci-type sequences: each term is the sum of the two previous terms (e.g. 1, 1, 2, 3, 5, 8, 13, ...).

Square numbers: 1, 4, 9, 16, 25, ... (positions $n^2$ ).

Cube numbers: 1, 8, 27, 64, 125, ... (positions $n^3$ ).

Triangular numbers: 1, 3, 6, 10, 15, ... (cumulative sum of natural numbers).

To continue a sequence: identify the pattern (difference, ratio, or rule), then apply it to find the next term(s).

Key termsgeometric sequencecommon ratioFibonacci sequencesquare numberstriangular numbers

Example

Identify the pattern and continue: 2, 6, 18, 54, ... The ratio is $6 ÷ 2 = 3$ . This is geometric with $r = 3$ . Next terms: $54 × 3 = 162$ and $162 × 3 = 486$ .

Think of it like this

A geometric sequence is like compound interest: each year your money is multiplied by the same factor. A Fibonacci sequence is like a rabbit population where each generation is the sum of the previous two generations.

Section 4

How do you find the nth term of a quadratic sequence? (Higher Tier)

A quadratic sequence has a constant second difference. The nth term formula has the form:

$a_n = An^2 + Bn + C$

Method:

Write out the sequence and calculate the first differences (differences between consecutive terms).
Calculate the second differences (differences between the first differences).
If the second differences are constant, the sequence is quadratic.
The second difference = $2A$ , so $A = \frac{\text{second difference}}{2}$ .
Use the formula and known terms to find $B$ and $C$ by substitution and solving simultaneous equations.

Example: 3, 8, 15, 24, 35, ...

First differences: 5, 7, 9, 11, ... (increasing by 2 each time)
Second difference: 2 (constant)
Therefore $A = 2 ÷ 2 = 1$
Using $a_1 = 3$ : $3 = 1(1)^2 + B(1) + C$ , so $B + C = 2$
Using $a_2 = 8$ : $8 = 1(4) + B(2) + C$ , so $2B + C = 4$
Solving: $B = 2$ , $C = 0$
Formula: $a_n = n^2 + 2n$

Key termsquadratic sequencefirst differencesecond difference

Exam tip

Always check that second differences are constant before assuming a sequence is quadratic. Examiners want to see your difference table clearly laid out and the formula checked against at least two terms.

Common mistake

Students sometimes forget to calculate the second difference correctly, or confuse it with the first difference. Lay out both rows clearly to avoid confusion.

Section 5

How do you work with Fibonacci and recursive sequences?

A recursive sequence (or recurrence relation) defines each term using one or more previous terms. The most famous example is the Fibonacci sequence:

$F_n = F_{n-1} + F_{n-2}$

starting with $F_1 = 1, F_2 = 1$ (or sometimes $F_0 = 0, F_1 = 1$ ).

The sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Other recursive sequences follow different rules. For example:

$a_n = 2a_{n-1} + 1$ with $a_1 = 3$ gives: 3, 7, 15, 31, 63, ...
$a_n = a_{n-1} + a_{n-2}$ with different starting values produces Fibonacci-type sequences.

To find terms in a recursive sequence:

Start with the given initial term(s).
Apply the recurrence relation repeatedly to find successive terms.
You must calculate all intermediate terms—you cannot jump directly to the nth term without working through the sequence.

Recursive sequences are useful for modelling real-world situations like population growth or financial savings.

Key termsrecursive sequencerecurrence relationFibonacci sequence

Example

Find the first five terms of the sequence defined by $a_n = 3a_{n-1} - 2$ with $a_1 = 4$ . Answer: $a_1 = 4$ , $a_2 = 3(4) - 2 = 10$ , $a_3 = 3(10) - 2 = 28$ , $a_4 = 3(28) - 2 = 82$ , $a_5 = 3(82) - 2 = 244$ . Sequence: 4, 10, 28, 82, 244.

Section 6

How do you determine if a value is a term in a sequence?

To check whether a given value is a term in a sequence, use the nth term formula and solve for $n$ .

For a linear sequence with formula $a_n = an + b$ :

Set $a_n = \text{given value}$ and solve for $n$ . If $n$ is a positive integer, the value is a term; otherwise, it is not.

Example: Is 50 a term in the sequence $a_n = 3n + 2$ ?

Set $3n + 2 = 50$
$3n = 48$
$n = 16$
Since $n = 16$ is a positive integer, 50 is a term in the sequence (it is the 16th term).

For quadratic sequences, substitute the given value into $a_n = An^2 + Bn + C$ and rearrange to form a quadratic equation. Use the discriminant or solve to check if $n$ is a positive integer.

For other sequences, you may need to check terms one by one or recognise the pattern. If the sequence is defined recursively, calculate terms until you either find the value or exceed it.

Key point: $n$ must always be a positive integer (1, 2, 3, ...) for a value to be in the sequence.

Key termsterm valuepositionpositive integer

Exam tip

Examiners want to see you set up the equation clearly and show your working. Always state whether n is or is not a positive integer in your conclusion, as this is what determines if the value is in the sequence.

Example

Is 100 a term in $a_n = 2n^2 - 3n + 1$ ? Set $2n^2 - 3n + 1 = 100$ , so $2n^2 - 3n - 99 = 0$ . Using the quadratic formula: $n = \frac{3 ± \sqrt{9 + 792}}{4} = \frac{3 ± \sqrt{801}}{4} ≈ 7.6$ or $n ≈ -6.6$ . Since n is not a positive integer, 100 is not a term in the sequence.

Must Know

Generate sequences using term-to-term rules (add/multiply to previous term) or position-to-term rules (formula for nth term).
Linear sequences have constant difference; the nth term is $a_n = an + b$ where $a$ is the common difference. Find $b$ by substitution.
Geometric sequences have constant ratio ( $a_n = ar^{n-1}$ ); recognise and continue by multiplying. Know square, cube, and triangular number sequences.
Quadratic sequences (HT) have constant second differences; the nth term is $a_n = An^2 + Bn + C$ . Find $A$ from second difference, then solve for $B$ and $C$ using simultaneous equations.
Fibonacci and recursive sequences define each term using previous terms (e.g. $F_n = F_{n-1} + F_{n-2}$ ); calculate all intermediate terms sequentially.
Test if a value is in a sequence by setting it equal to the nth term formula and solving for $n$ —the value is in the sequence only if $n$ is a positive integer.

That's the notes covered.

Carry on to the next subtopic.