A23, A24, A25AlgebraFoundation & HigherGrades 2–7

Sequences

A sequence is an ordered list of numbers that follows a rule. A term-to-term rule tells you how to get from one term to the next; an $n$ th term rule tells you any term directly from its position. The $n$ th term is the powerful one: it answers "what is the 100th term?" and "is 250 in the sequence?" without listing anything.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

A linear sequence goes up or down by the same amount each time. That constant difference is the multiplier of $n$ in the $n$ th term — the sequence $5, 8, 11, 14, \ldots$ climbs in 3s, so it is "the 3 times table, shifted": $3n + 2$ .

A quadratic sequence has a constant $\textbf{second}$ difference, and its $n$ th term includes $n^2$ . A geometric sequence multiplies by the same number each time. Recognising which kind you have is the first step every time.

Sequences and Finding the Nth TermCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Continue sequences using term-to-term rules
Recognise triangular, square, cube, Fibonacci-type and geometric sequences
Find and use the $n$ th term of a linear sequence
Find the $n$ th term of a quadratic sequence (Higher)

Term-to-term and special sequences

A term-to-term rule describes how each term follows the one before, such as "add 4" or "multiply by 3".

Learn these: square numbers $1, 4, 9, 16, \ldots$ ; cube numbers $1, 8, 27, 64, \ldots$ ; triangular numbers $1, 3, 6, 10, \ldots$ ; Fibonacci-type sequences, where each term is the sum of the previous two, such as $1, 1, 2, 3, 5, 8, \ldots$ ; and geometric sequences, which multiply by a common ratio, such as $3, 6, 12, 24, \ldots$

Example 1A Fibonacci-type sequence starts

2, 5, 7, 12, \ldots

Find the next two terms.

1Each term is the sum of the previous two:

7 + 12 = 19

12 + 19 = 31

Answer

19, 31

Example 2A geometric sequence starts

4, 12, 36, \ldots

Find the 6th term.

1The common ratio is

12 \div 4 = 3

2Terms:

4, 12, 36, 108, 324, 972

Answer

972

Your turnSame method, new numbers — try it before moving on.

A geometric sequence starts $800, 400, 200, \ldots$ Find the 6th term.

Your answer

Show worked solution

Solution

1The common ratio is $\frac{1}{2}$ .
2 $800, 400, 200, 100, 50, 25$ .

Answer: $25$

The nth term of a linear sequence

Find the common difference $d$ — that is the coefficient of $n$ . Then compare the sequence with $dn$ to find the adjustment.

For $5, 8, 11, 14, \ldots$ : the difference is 3, so compare with $3n = 3, 6, 9, 12, \ldots$ . Each term is 2 more, so the $n$ th term is $3n + 2$ .

Example 1Find the

n

th term of

7, 3, -1, -5, \ldots

1The difference is

-4

, so compare with

-4n

-4, -8, -12, -16

2Each term is 11 more than that.

Answer

-4n + 11

(or

11 - 4n

)

Your turnSame method, new numbers — try it before moving on.

Find the $n$ th term of $4, 9, 14, 19, \ldots$

Your answer

Type it like 3n + 2.

Show worked solution

Solution

1The difference is 5: compare with $5n = 5, 10, 15, 20$ .
2Each term is 1 less: $5n - 1$ .

Answer: $5n - 1$

Example 2Find the

n

th term of

20, 17, 14, 11, \ldots

1Difference

-3

-3n = -3, -6, -9, -12

2Each term is 23 more.

Answer

23 - 3n

Your turnSame method, new numbers — try it before moving on.

Find the $n$ th term of $15, 13, 11, 9, \ldots$

Your answer

Type it like 20 - 3n.

Show worked solution

Solution

1Difference $-2$ : $-2n = -2, -4, -6, -8$ .
2Each term is 17 more: $17 - 2n$ .

Answer: $17 - 2n$

Tip — Check your rule with $n = 1$ and $n = 2$ — they must give the first two terms.

Using the nth term

Substitute a position to find a term: the 50th term of $3n + 2$ is $3(50) + 2 = 152$ .

To test whether a number is in the sequence, set the $n$ th term equal to it and solve. It is in the sequence only if $n$ is a positive whole number.

Example 1Is 100 a term of the sequence with

n

th term

3n + 2

1Solve

3n + 2 = 100

3n = 98

, so

n = 32.67\ldots

n

is not a whole number.

Answer

No — 100 is not in the sequence.

Your turnSame method, new numbers — try it before moving on.

The $n$ th term of a sequence is $6n - 5$ . Which term is equal to 133?

Your answer

Show worked solution

Solution

1 $6n - 5 = 133$ , so $6n = 138$ .
2 $n = 23$ .

Answer: The 23rd term

Explaining without solving

Every term of $3n + 2$ is 2 more than a multiple of 3. $100 = 99 + 1$ is 1 more than a multiple of 3, so it cannot be in the sequence. Reasoning like this is exactly what "explain why" questions want.

Quadratic sequences (Higher)

If the first differences change but the second differences are constant, the sequence is quadratic. The coefficient of $n^2$ is $\textbf{half}$ the second difference.

Subtract that $n^2$ part from each term. What is left is a linear sequence — find its $n$ th term and add it on.

Example 1Find the

n

th term of

4, 11, 22, 37, 56, \ldots

1First differences:

7, 11, 15, 19

. Second differences:

4

. So the

n^2

coefficient is 2.

2Subtract

2n^2 = 2, 8, 18, 32, 50

: leaves

2, 3, 4, 5, 6

3That remainder is

n + 1

Answer

2n^2 + n + 1

Your turnSame method, new numbers — try it before moving on.

Find the $n$ th term of $2, 7, 14, 23, 34, \ldots$

Your answer

Type n² as n^2.

Show hint

The second difference is 2, so start with $n^2$ .

Show worked solution

Solution

1Differences $5, 7, 9, 11$ ; second difference 2, so $n^2$ .
2Subtract $n^2 = 1, 4, 9, 16, 25$ : leaves $1, 3, 5, 7, 9$ .
3That is $2n - 1$ , so the $n$ th term is $n^2 + 2n - 1$ .

Answer: $n^2 + 2n - 1$

Think like an examiner

"Is 200 in the sequence? Give a reason" needs working: solve the equation, or explain using what every term has in common. "No" alone scores nothing.
Write the $n$ th term as an expression, not " $n + 3$ each time" — that is a term-to-term rule.
In quadratic sequences, show the second differences and the subtracted sequence; method marks are awarded for each.

Rules

u_n = dn + (a - d)

Linear:

a

is the first term,

d

the difference.

n^2 \text{ coefficient} = \tfrac{1}{2} \times \text{second difference}

Quadratic sequences.

u_{n+1} = r \times u_n

Geometric: common ratio

r

Watch out for these

The

n

th term of

5, 8, 11, \ldots

n + 3

"Add 3" is the term-to-term rule. The

n

th term is

3n + 2

Using

n^2

coefficient = second difference:

4n^2

for a second difference of 4.

Halve it:

2n^2

For a decreasing sequence, writing

3n + 23

instead of

-3n + 23

A sequence going down has a negative coefficient of

n

Saying 100 is the 32.67th term.

Positions are whole numbers, so 100 is not a term.

Stretch yourself

Sequence A has $n$ th term $4n + 1$ . Sequence B has $n$ th term $7n - 5$ . Find the three smallest numbers that appear in both sequences.

Hint — List some terms of each, then look for a pattern in the common terms.

Reveal solution

SolutionA:

5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, \ldots

2, 9, 16, 23, 30, 37, 44, 51, 58, 65, \ldots

Common terms:

9, 37, 65

. They increase by 28, the lowest common multiple of 4 and 7.

\blacksquare

Questions students ask

What is the difference between arithmetic and geometric sequences?

Arithmetic (linear) sequences add the same amount each time. Geometric sequences multiply by the same amount each time.

Do I need to know the

n

th term of a geometric sequence?

At GCSE you should be able to continue geometric sequences and find terms by repeated multiplication. Some Higher questions use

ar^{n-1}

, but this is not required in general.

How do I know if a sequence is quadratic?

Work out the differences between terms. If those differences change, find the differences of the differences. If the second differences are all the same, it is quadratic.

Key takeaways

Term-to-term rules link consecutive terms; $n$ th terms give any term directly.
Linear $n$ th term: common difference times $n$ , plus an adjustment.
To test if a number is a term, solve and check $n$ is a positive whole number.
Quadratic: $n^2$ coefficient is half the second difference.
Know square, cube, triangular, Fibonacci-type and geometric sequences.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 33 marks

Here are the first four terms of a sequence: $\quad 6, \ 11, \ 16, \ 21$

(a)Write down an expression for the $n$ th term.[2]
Show mark scheme (a)
- B1 $5n + k$ for any $k \ne 1$
- B1 $5n + 1$
Answer: $5n + 1$
(b)Is 83 a term of the sequence? Give a reason for your answer.[1]
Show mark scheme (b)
- B1No, with a reason, e.g. $5n + 1 = 83$ gives $n = 16.4$ , not a whole number; or every term ends in 1 or 6
Answer: No

Question 2Grade 53 marks

The $n$ th term of a sequence is $2n^2 + 3$ .

(a)Work out the 5th term.[1]
Show mark scheme (a)
- B1 $53$
Answer: $53$
(b)Is 131 a term of the sequence? Show how you decide.[2]
Show mark scheme (b)
- M1Forms $2n^2 + 3 = 131$ and rearranges to $n^2 = 64$
- A1Yes — $n = 8$ , so 131 is the 8th term
Answer: Yes, the 8th term

Examiner note

Take only the positive square root: positions in a sequence are positive whole numbers.

Question 3Grade 73 marks

Find an expression for the $n$ th term of the quadratic sequence $\quad 3, \ 10, \ 21, \ 36, \ 55$ [3]
Show mark scheme
- M1Second differences of 4 found, or $2n^2$ used
- M1Subtracts $2n^2$ to give the linear sequence $1, 2, 3, 4, 5$
- A1 $2n^2 + n$
Answer: $2n^2 + n$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

What is the next term of $3, 7, 11, 15, \ldots$ ?
- $19$
- $18$
- $20$
- $16$
Show answer and explanation
Answer: $19$
Add 4 each time.
Which of these is a sequence of triangular numbers?
- $1, 3, 6, 10$
- $1, 4, 9, 16$
- $1, 2, 4, 8$
- $1, 1, 2, 3$
Show answer and explanation
Answer: $1, 3, 6, 10$
Add 2, then 3, then 4, …
Find the $n$ th term of $2, 5, 8, 11, \ldots$
- $3n - 1$
- $n + 3$
- $3n + 2$
- $2n + 3$
Show answer and explanation
Answer: $3n - 1$
Compare with $3n$ : each term is 1 less.
The $n$ th term is $4n - 7$ . What is the 20th term?
- $73$
- $80$
- $13$
- $87$
Show answer and explanation
Answer: $73$
$4 \times 20 - 7$ .
Find the $n$ th term of $12, 9, 6, 3, \ldots$
- $15 - 3n$
- $3n + 9$
- $12 - 3n$
- $-3n + 12$
Show answer and explanation
Answer: $15 - 3n$
Compare with $-3n$ : each term is 15 more.
What is the 5th term of the geometric sequence $2, 6, 18, \ldots$ ?
- $162$
- $54$
- $486$
- $30$
Show answer and explanation
Answer: $162$
Multiply by 3: $2, 6, 18, 54, 162$ .
A quadratic sequence has second difference 6. What is the coefficient of $n^2$ ?
- $3$
- $6$
- $12$
- $2$
Show answer and explanation
Answer: $3$
Half the second difference.
Find the $n$ th term of $0, 3, 8, 15, 24, \ldots$
- $n^2 - 1$
- $n^2 + 1$
- $2n^2 - 2$
- $3n - 3$
Show answer and explanation
Answer: $n^2 - 1$
Each term is one less than a square number.

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