1MA1 · A23–A25AlgebraFoundation & HigherGrades 3–7

Sequences

A sequence is a list of numbers that follows a rule. Some rules describe how to get from one term to the next; better rules give any term directly from its position, so you can find the 100th term without writing out the first 99. That position-to-term rule is the $n$ th term, and finding it is the core skill of this topic.

50 min Sequences & Functions

The big picture

Differences reveal the type of sequence. If the first differences are constant, the sequence is linear and its $n$ th term is $an + b$ . If the second differences are constant, it is quadratic, $an^{2} + bn + c$ . If each term is multiplied by the same number, it is geometric.

Edexcel regularly asks whether a number is a term of a sequence — which is really an equation to solve for $n$ , where the answer must be a positive whole number — and at Higher tier asks for the $n$ th term of a quadratic sequence.

What you'll learn

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

Types of rule

A $\textbf{term-to-term}$ rule tells you how to get the next term: "start at 4, add 3" gives $4, 7, 10, 13, \dots$

A $\textbf{position-to-term}$ rule, or $n$ th term, gives the value from its position $n$ : $3n + 1$ gives $4, 7, 10, 13, \dots$ for $n = 1, 2, 3, 4$ .

Special sequences to recognise: square numbers $1, 4, 9, 16$ ; cube numbers $1, 8, 27, 64$ ; triangular numbers $1, 3, 6, 10$ .

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

Find the next term of the geometric sequence $2, 6, 18, 54, \ldots$

Your answer

Show worked solution

Solution

1The common ratio is 3.
2 $54 \times 3 = 162$ .

Answer: $162$

Linear sequences

A linear (arithmetic) sequence has a constant difference $d$ . Its $n$ th term is $dn + b$ , where $b$ is found by comparing with the sequence $dn$ .

For $7, 11, 15, 19$ : the difference is 4, so start with $4n$ ( $4, 8, 12, 16$ ). Each term is 3 more, so the $n$ th term is $4n + 3$ .

To test whether a number is in the sequence, set the $n$ th term equal to it and solve. A whole-number $n$ means yes.

Example 1Is 150 a term of the sequence

7, 11, 15, 19, \dots

n

th term:

4n + 3

2Solve

4n + 3 = 150

4n = 147

, so

n = 36.75

AnswerNo —

n

is not a whole number, so 150 is not in the sequence.

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

Find the $n$ th term of $5, 12, 19, 26, \ldots$

Your answer

Type it like 4n + 1.

Show worked solution

Solution

1The difference is 7: compare with $7n = 7, 14, 21$ .
2Each term is 2 less: $7n - 2$ .

Answer: $7n - 2$

Tip — Check the $n$ th term by substituting $n = 1$ and $n = 2$ . It takes seconds and catches wrong constants.

Geometric and Fibonacci-type sequences

A $\textbf{geometric}$ sequence multiplies by the same number $r$ each time: $5, 15, 45, 135, \dots$ has $r = 3$ . Its $n$ th term is $ar^{n-1}$ , here $5 \times 3^{n - 1}$ .

A $\textbf{Fibonacci-type}$ sequence adds the two previous terms: starting $2, 5$ gives $7, 12, 19, 31, \dots$

Geometric ratios can be fractions or surds: $\sqrt2, 2, 2\sqrt2, 4$ has ratio $\sqrt2$ .

Example 1A Fibonacci-type sequence begins

a, b, a + b, a + 2b, \dots

The 3rd term is 11 and the 5th term is 29. Find

a

and

b

1Terms:

a

b

a + b

a + 2b

2a + 3b

a + b = 11

and

2a + 3b = 29

3Doubling the first:

2a + 2b = 22

; subtract from the second:

b = 7

, so

a = 4

Answer

a = 4

b = 7

Quadratic sequences

If the first differences change but the $\textbf{second}$ differences are constant, the sequence is quadratic.

The coefficient of $n^{2}$ is half the second difference. Subtract that $n^{2}$ term from the sequence, and what remains is linear — find its $n$ th term as usual.

Example 1Find the

n

th term of

3, 8, 15, 24, 35, \dots

1First differences:

5, 7, 9, 11

. Second differences:

2

. So the

n^{2}

coefficient is 1.

2Subtract

n^{2}

(

1, 4, 9, 16, 25

): remainder

2, 4, 6, 8, 10

3The remainder has

n

th term

2n

Answer

n^{2} + 2n

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

Find the $n$ th term of $4, 10, 18, 28, 40, \ldots$

Your answer

Type n² as n^2.

Show worked solution

Solution

1Differences $6, 8, 10, 12$ ; second difference 2, so $n^2$ .
2Subtract $n^2$ : $3, 6, 9, 12, 15 = 3n$ . So $n^2 + 3n$ .

Answer: $n^2 + 3n$

Why this works

Why half the second difference? For $an^{2}$ alone, consecutive terms $a, 4a, 9a, 16a$ have differences $3a, 5a, 7a$ and second difference $2a$ . So a second difference of 2 means $a = 1$ .

Think like an examiner

Show the differences you calculated.
When deciding if a number is a term, show the equation and interpret $n$ .
For quadratic sequences, show the subtraction of the $an^{2}$ sequence.

Sequences

dn + b

Linear nth term, difference d.

ar^{n-1}

Geometric nth term.

n^{2} \text{ coefficient} = \tfrac12 \times \text{second difference}

Quadratic sequences.

Watch out for these

Writing the

n

th term of

7, 11, 15

n + 4

The difference multiplies

n

4n + 3

Using the second difference itself as the

n^{2}

coefficient.

Halve it.

Saying 150 is term 36.75.

Positions must be whole numbers, so it is not a term.

Treating

2, 6, 18, 54

as linear.

It multiplies by 3: geometric.

Stretch yourself

Find the $n$ th term of the quadratic sequence $2, 9, 20, 35, 54, \dots$ and hence decide whether 170 is a term.

Hint — Second differences first. Then set your $n$ th term equal to 170 and solve the quadratic.

Reveal solution

SolutionFirst differences:

7, 11, 15, 19

; second differences:

4

, so the

n^{2}

coefficient is 2. Subtracting

2n^{2}

(

2, 8, 18, 32, 50

) leaves

0, 1, 2, 3, 4

, which is

n - 1

. So the

n

th term is

2n^{2} + n - 1

. For 170:

2n^{2} + n - 171 = 0

, which factorises as

(2n + 19)(n - 9) = 0

, so

n = 9

. Yes, 170 is the 9th term. Check:

2(81) + 9 - 1 = 170

✓.

\blacksquare

Questions students ask

How do I know if a sequence is quadratic?

If the first differences are not constant but the second differences are.

What is the difference between arithmetic and geometric?

Arithmetic sequences add a constant; geometric sequences multiply by a constant.

Can a sequence decrease?

Yes. A negative common difference gives a decreasing linear sequence, such as

20, 17, 14

with

n

th term

23 - 3n

Key takeaways

Term-to-term rules link consecutive terms; the $n$ th term gives any term directly.
Linear: constant difference $d$ , $n$ th term $dn + b$ .
Geometric: constant ratio $r$ , $n$ th term $ar^{n - 1}$ .
Quadratic: constant second difference; $n^{2}$ coefficient is half of it.
To test membership, solve for $n$ and check it is a positive integer.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 32 marks

Find an expression for the $n$ th term of $20, 17, 14, 11, \ldots$ [2]
Show mark scheme
- M1 $-3n + k$
- A1 $23 - 3n$
Answer: $23 - 3n$

Question 2Grade 52 marks

A geometric sequence starts $3, 12, 48, \ldots$ Work out the 6th term.[2]
Show mark scheme
- M1Common ratio 4
- A1 $3072$
Answer: $3072$

Question 3Grade 73 marks

Find an expression for the $n$ th term of the quadratic sequence $5, 11, 19, 29, 41, \ldots$ [3]
Show mark scheme
- M1Second difference 2, so $n^2$
- M1Remaining sequence $4, 7, 10, 13, 16$
- A1 $n^2 + 3n + 1$
Answer: $n^2 + 3n + 1$

Unlimited practice

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