AlgebraAQA GCSE Mathematics (8300)

Sequences

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next. (2026)

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Worked examples4 Key formulae4 Definitions6 Common mistakes4

Step-by-step worked examples

01.Finding the nth term of a linear sequence
Foundation
Question
Find the nth term of the sequence $8, 13, 18, 23, 28, \ldots$
Solution
1. 1
  Find the common difference by subtracting consecutive terms.
  $d = 13 - 8 = 5$
2. 2
  The coefficient of $n$ is $d = 5$ . Find the constant by subtracting $d$ from the first term (the 'zeroth term').
  $c = 8 - 5 = 3$
3. 3
  Write the nth term formula.
  $T_n = 5n + 3$
4. 4
  Verify with $n = 1$ and $n = 5$ .
  $T_1 = 5(1) + 3 = 8 \checkmark; \quad T_5 = 5(5) + 3 = 28 \checkmark$
Answer
$T_n = 5n + 3$
02.Finding the nth term of a quadratic sequence
Higher
Question
Find the nth term of the sequence $4, 11, 22, 37, 56, \ldots$
Solution
1. 1
  Calculate first differences.
  $7, \; 11, \; 15, \; 19$
2. 2
  Calculate second differences — these are constant.
  $4, \; 4, \; 4$
3. 3
  Find $a$ : half the second difference.
  $a = 4 \div 2 = 2$
4. 4
  Subtract $2n^2$ from each term to find the linear remainder.
  $4-2=2, \; 11-8=3, \; 22-18=4, \; 37-32=5, \; 56-50=6$
5. 5
  The remainder sequence $2, 3, 4, 5, 6, \ldots$ has nth term $n + 1$ . Combine.
  $T_n = 2n^2 + n + 1$
Answer
$T_n = 2n^2 + n + 1$ ; check: $T_1 = 2+1+1 = 4$ ✓, $T_5 = 50+5+1 = 56$ ✓
03.Determining whether a value is a term of a sequence
Higher
Question
The nth term of a sequence is $n^2 + 3n$ . Is 130 a term of the sequence?
Solution
1. 1
  Set the nth term equal to 130.
  $n^2 + 3n = 130$
2. 2
  Rearrange to standard form.
  $n^2 + 3n - 130 = 0$
3. 3
  Factorise.
  $(n + 13)(n - 10) = 0$
4. 4
  Solve and interpret.
  $n = 10 \text{ or } n = -13$
5. 5
  $n = 10$ is a positive integer, so 130 is a term of the sequence.
Answer
Yes — 130 is the 10th term of the sequence.
04.Working with a geometric sequence
Higher
Question
A geometric sequence has first term 4 and third term 36. Find the common ratio and the 6th term.
Solution
1. 1
  Use the nth term formula for the third term: $T_3 = 4r^2 = 36$ .
  $r^2 = 9 \implies r = 3 \; (\text{taking the positive root})$
2. 2
  Write out the sequence: $4, 12, 36, 108, 324, 972, \ldots$
3. 3
  Find the 6th term using the formula.
  $T_6 = 4 \times 3^{5} = 4 \times 243 = 972$
Answer
Common ratio $r = 3$ ; 6th term $= 972$ .

Key formulae

nth Term of an Arithmetic Sequence
$T_n = a + (n-1)d$
- $a$
  — first term
- $d$
  — common difference
- $n$
  — term position (positive integer)
When to use
Finding any term of an arithmetic sequence; also written as $T_n = dn + (a - d)$ to read off the coefficient and constant directly.
nth Term of a Geometric Sequence
$T_n = ar^{n-1}$
- $a$
  — first term
- $r$
  — common ratio
- $n$
  — term position
When to use
Finding a specific term or the general formula for a geometric (exponential) sequence.
nth Triangular Number
$T_n = \frac{n(n+1)}{2}$
- $n$
  — term position
When to use
Generating or checking triangular numbers; recognising when a sequence fits the triangular number pattern.
Coefficient of $n^2$ in a Quadratic Sequence
$a = \frac{\text{second difference}}{2}$
- $a$
  — coefficient of $n^2$ in the nth term formula
- $\text{second difference}$
  — constant value of $\Delta^2 T_n$
When to use
First step in finding the nth term of any quadratic sequence — always halve the second difference to get the $n^2$ coefficient.

Practice with flashcards

Card 1 of 60 known · 0 to review

Key definitions and keywords

Arithmetic sequence: A sequence in which each term is obtained by adding (or subtracting) a constant value, called the common difference, to the previous term.
Geometric sequence: A sequence in which each term is obtained by multiplying the previous term by a constant value called the common ratio.
nth term (position-to-term rule): A formula that gives the value of any term in a sequence directly from its position number $n$ , without needing to know the previous term.
Second difference: The difference between consecutive first differences. A constant second difference indicates that the sequence is quadratic (the nth term contains an $n^2$ term).
Triangular numbers: The sequence $1, 3, 6, 10, 15, \ldots$ formed by summing consecutive natural numbers. The $n$ th triangular number is $\frac{n(n+1)}{2}$ .
Fibonacci sequence: A sequence where each term is the sum of the two preceding terms, starting from 1, 1: $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ The term-to-term rule is $u_{n+2} = u_{n+1} + u_n$ .

Common mistakes and misconceptions

Mistake
Using the first difference instead of half the second difference as the coefficient of $n^2$
Why it happens
Students see the differences table and mistake the first difference for the multiplier of $n^2$ , forgetting that a quadratic grows faster than a linear sequence.
How to avoid it
Memorise: coefficient of $n^2 =$ second difference $\div 2$ . After finding $a$ , subtract $an^2$ and check that the remainder is linear — if it is not, recheck the second difference.
Mistake
Finding the 0th term instead of the constant in a linear nth term
Why it happens
Students set $n = 0$ in their part-completed formula to find the constant $c$ , but haven't finished the formula yet — this circular reasoning leads to errors.
How to avoid it
Use the explicit method: constant $c = a - d$ (first term minus common difference). Then write $T_n = dn + c$ and verify with $n = 1$ .
Mistake
Failing to pair each $x$ value with the correct $y$ value when checking if a value is in a quadratic sequence
Why it happens
After solving the quadratic equation, students forget to check both solutions and accept a negative or non-integer value of $n$ .
How to avoid it
Explicitly state: 'For $n$ to represent a term position, it must be a positive integer.' Reject any non-integer or non-positive solution and state the conclusion clearly.
Mistake
Confusing arithmetic and geometric sequences by assuming all sequences are arithmetic
Why it happens
Students default to finding differences even when the sequence is clearly multiplicative, leading to non-constant differences and confusion.
How to avoid it
Before calculating differences, check whether the ratio between consecutive terms is constant. If both the difference and ratio are constant, the sequence is trivially constant (ratio 1, difference 0).

AlgebraAQA GCSE Mathematics (8300)

Sequences

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next. (2026)

PreviousNext

Master the topic

Worked examples4 Key formulae4 Definitions6 Common mistakes4

Step-by-step worked examples

01.Finding the nth term of a linear sequence
Foundation
Question
Find the nth term of the sequence $8, 13, 18, 23, 28, \ldots$
Solution
1. 1
  Find the common difference by subtracting consecutive terms.
  $d = 13 - 8 = 5$
2. 2
  The coefficient of $n$ is $d = 5$ . Find the constant by subtracting $d$ from the first term (the 'zeroth term').
  $c = 8 - 5 = 3$
3. 3
  Write the nth term formula.
  $T_n = 5n + 3$
4. 4
  Verify with $n = 1$ and $n = 5$ .
  $T_1 = 5(1) + 3 = 8 \checkmark; \quad T_5 = 5(5) + 3 = 28 \checkmark$
Answer
$T_n = 5n + 3$
02.Finding the nth term of a quadratic sequence
Higher
Question
Find the nth term of the sequence $4, 11, 22, 37, 56, \ldots$
Solution
1. 1
  Calculate first differences.
  $7, \; 11, \; 15, \; 19$
2. 2
  Calculate second differences — these are constant.
  $4, \; 4, \; 4$
3. 3
  Find $a$ : half the second difference.
  $a = 4 \div 2 = 2$
4. 4
  Subtract $2n^2$ from each term to find the linear remainder.
  $4-2=2, \; 11-8=3, \; 22-18=4, \; 37-32=5, \; 56-50=6$
5. 5
  The remainder sequence $2, 3, 4, 5, 6, \ldots$ has nth term $n + 1$ . Combine.
  $T_n = 2n^2 + n + 1$
Answer
$T_n = 2n^2 + n + 1$ ; check: $T_1 = 2+1+1 = 4$ ✓, $T_5 = 50+5+1 = 56$ ✓
03.Determining whether a value is a term of a sequence
Higher
Question
The nth term of a sequence is $n^2 + 3n$ . Is 130 a term of the sequence?
Solution
1. 1
  Set the nth term equal to 130.
  $n^2 + 3n = 130$
2. 2
  Rearrange to standard form.
  $n^2 + 3n - 130 = 0$
3. 3
  Factorise.
  $(n + 13)(n - 10) = 0$
4. 4
  Solve and interpret.
  $n = 10 \text{ or } n = -13$
5. 5
  $n = 10$ is a positive integer, so 130 is a term of the sequence.
Answer
Yes — 130 is the 10th term of the sequence.
04.Working with a geometric sequence
Higher
Question
A geometric sequence has first term 4 and third term 36. Find the common ratio and the 6th term.
Solution
1. 1
  Use the nth term formula for the third term: $T_3 = 4r^2 = 36$ .
  $r^2 = 9 \implies r = 3 \; (\text{taking the positive root})$
2. 2
  Write out the sequence: $4, 12, 36, 108, 324, 972, \ldots$
3. 3
  Find the 6th term using the formula.
  $T_6 = 4 \times 3^{5} = 4 \times 243 = 972$
Answer
Common ratio $r = 3$ ; 6th term $= 972$ .

Key formulae

nth Term of an Arithmetic Sequence
$T_n = a + (n-1)d$
- $a$
  — first term
- $d$
  — common difference
- $n$
  — term position (positive integer)
When to use
Finding any term of an arithmetic sequence; also written as $T_n = dn + (a - d)$ to read off the coefficient and constant directly.
nth Term of a Geometric Sequence
$T_n = ar^{n-1}$
- $a$
  — first term
- $r$
  — common ratio
- $n$
  — term position
When to use
Finding a specific term or the general formula for a geometric (exponential) sequence.
nth Triangular Number
$T_n = \frac{n(n+1)}{2}$
- $n$
  — term position
When to use
Generating or checking triangular numbers; recognising when a sequence fits the triangular number pattern.
Coefficient of $n^2$ in a Quadratic Sequence
$a = \frac{\text{second difference}}{2}$
- $a$
  — coefficient of $n^2$ in the nth term formula
- $\text{second difference}$
  — constant value of $\Delta^2 T_n$
When to use
First step in finding the nth term of any quadratic sequence — always halve the second difference to get the $n^2$ coefficient.

Practice with flashcards

Card 1 of 60 known · 0 to review

Key definitions and keywords

Arithmetic sequence: A sequence in which each term is obtained by adding (or subtracting) a constant value, called the common difference, to the previous term.
Geometric sequence: A sequence in which each term is obtained by multiplying the previous term by a constant value called the common ratio.
nth term (position-to-term rule): A formula that gives the value of any term in a sequence directly from its position number $n$ , without needing to know the previous term.
Second difference: The difference between consecutive first differences. A constant second difference indicates that the sequence is quadratic (the nth term contains an $n^2$ term).
Triangular numbers: The sequence $1, 3, 6, 10, 15, \ldots$ formed by summing consecutive natural numbers. The $n$ th triangular number is $\frac{n(n+1)}{2}$ .
Fibonacci sequence: A sequence where each term is the sum of the two preceding terms, starting from 1, 1: $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ The term-to-term rule is $u_{n+2} = u_{n+1} + u_n$ .

Common mistakes and misconceptions

Mistake
Using the first difference instead of half the second difference as the coefficient of $n^2$
Why it happens
Students see the differences table and mistake the first difference for the multiplier of $n^2$ , forgetting that a quadratic grows faster than a linear sequence.
How to avoid it
Memorise: coefficient of $n^2 =$ second difference $\div 2$ . After finding $a$ , subtract $an^2$ and check that the remainder is linear — if it is not, recheck the second difference.
Mistake
Finding the 0th term instead of the constant in a linear nth term
Why it happens
Students set $n = 0$ in their part-completed formula to find the constant $c$ , but haven't finished the formula yet — this circular reasoning leads to errors.
How to avoid it
Use the explicit method: constant $c = a - d$ (first term minus common difference). Then write $T_n = dn + c$ and verify with $n = 1$ .
Mistake
Failing to pair each $x$ value with the correct $y$ value when checking if a value is in a quadratic sequence
Why it happens
After solving the quadratic equation, students forget to check both solutions and accept a negative or non-integer value of $n$ .
How to avoid it
Explicitly state: 'For $n$ to represent a term position, it must be a positive integer.' Reject any non-integer or non-positive solution and state the conclusion clearly.
Mistake
Confusing arithmetic and geometric sequences by assuming all sequences are arithmetic
Why it happens
Students default to finding differences even when the sequence is clearly multiplicative, leading to non-constant differences and confusion.
How to avoid it
Before calculating differences, check whether the ratio between consecutive terms is constant. If both the difference and ratio are constant, the sequence is trivially constant (ratio 1, difference 0).

Sequences

Master the topic

Step-by-step worked examples

01.Finding the nth term of a linear sequence

02.Finding the nth term of a quadratic sequence

03.Determining whether a value is a term of a sequence

04.Working with a geometric sequence

Key formulae

Practice with flashcards

Key definitions and keywords

Common mistakes and misconceptions

Sequences

Master the topic

Step-by-step worked examples

01.Finding the nth term of a linear sequence

02.Finding the nth term of a quadratic sequence

03.Determining whether a value is a term of a sequence

04.Working with a geometric sequence

Key formulae

Practice with flashcards

Key definitions and keywords

Common mistakes and misconceptions