Sequences

Syllabus coverage

Continue a given number sequence or pattern
Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences
Find and use the $n$ th term of linear, simple quadratic and simple cubic sequences
Extended only: subscript notation may be used, e.g. $T_{n}$ is the $n$ th term of sequence T
Extended only: includes linear, quadratic, cubic and exponential sequences and simple combinations of these

The sequence 7, 11, 15, 19 with its first differences, and the same terms plotted on the line Tₙ = 4n + 3.

Term-to-term rules: what comes next

A sequence is a list of numbers in a definite order, and each number in it is a term. The order carries meaning: 1, 4, 9, 16 is a different sequence from 16, 9, 4, 1.

A term-to-term rule says how to get from one term to the next. In 2, 6, 12, 20 the gaps are 4, 6, 8, 10, so the next gap is 12 and the next two terms are 30 and 42. Writing the differences underneath the terms is the first move in almost every sequence question, and it takes about ten seconds.

A rule can be described in words as well as in numbers: 3, 6, 12, 24 doubles each time, so the next two terms are 48 and 96. 'Multiply by 2' is as much a term-to-term rule as 'add 3' is.

Extended candidates may meet the rule written with subscripts. $T_{n}$ means the $n$ th term of sequence T, so $T_{1}$ is the first term and $T_{5}$ the fifth. A rule such as $T_{n + 1} = 2 T_{n} + 1$ reads: take a term, double it, add 1, and you have the next term. Starting from $T_{1} = 2$ that gives 2, 5, 11, 23, ...

Sequence	Differences	Term-to-term rule	Next two terms
2, 6, 12, 20	4, 6, 8	add the next even number	30, 42
3, 6, 12, 24	3, 6, 12 - not constant	multiply by 2	48, 96
20, 17, 14, 11	-3 each time	subtract 3	8, 5
2, 5, 11, 23	3, 6, 12	double and add 1	47, 95

Write the differences under the terms before you do anything else. A constant difference, a constant ratio or a constant second difference tells you immediately which family of sequence you are looking at.

Position-to-term rules and the nth term

A position-to-term rule connects a term to its position in the list instead of to the term before it. The positions are numbered 1, 2, 3, 4, ... and the rule is a formula in $n$ . For the sequence 5, 8, 11, 14 the rule is $T_{n} = 3 n + 2$ : put in the position and it gives out the term.

The $n$ th term is the general term, the formula that would produce every term of the sequence. Once you have it, the 50th term is one substitution: $3 \times 50 + 2 = 152$ . That is why examiners ask for it, because counting on by hand is not the skill being tested.

The two kinds of rule do different jobs. A term-to-term rule continues a sequence one step at a time; a position-to-term rule jumps straight to any term you like. A question that asks for the 20th term or the 100th term is asking for the position-to-term rule.

Positions 1, 2, 3, 4 give the terms 5, 8, 11, 14
Position-to-term rule: $T_{n} = 3 n + 2$
$T_{20} = 3 \times 20 + 2 = 62$
$T_{n}$ is a formula, while $T_{4}$ is a single term

A rule drawn as a machine: one input gives one output. The nth term works the same way, with the position n going in and the term coming out.

Finding the nth term of a linear sequence

A linear sequence has a constant difference between consecutive terms, and its nth term always looks like $T_{n} = a n + b$ . The common difference is the coefficient of $n$ : in 7, 11, 15, 19 the difference is 4, so the rule starts $4 n$ .

The 4 in $4 n$ is not the whole answer, because $4 \times 1 = 4$ while the first term is 7. Find what has to be added by substituting $n = 1$ : $4 (1) + b = 7$ , so $b = 3$ and $T_{n} = 4 n + 3$ . Check it on the second term: $4 (2) + 3 = 11$ .

A decreasing sequence has a negative common difference, and the algebra copes with it. In 20, 17, 14, 11 the difference is -3, so the rule starts $- 3 n$ . Substituting $n = 1$ : $- 3 + b = 20$ , so $b = 23$ and $T_{n} = 23 - 3 n$ . Mark schemes accept $23 - 3 n$ and $- 3 n + 23$ equally; they are the same expression.

Always check with $n = 1$ and $n = 2$ before you move on. Two substitutions take five seconds and catch every slipped constant.

7, 11, 15, 19: difference 4, so $T_{n} = 4 n + 3$
20, 17, 14, 11: difference -3, so $T_{n} = 23 - 3 n$
3, 7, 11, 15: difference 4, so $T_{n} = 4 n - 1$
Check: $T_{1}$ must be the first term and $T_{2}$ the second

The coefficient of

n

is the common difference, and the constant is whatever makes

n = 1

give the first term. Two steps, and each one is worth a mark.

Quadratic sequences: when the second difference is constant

In 3, 6, 11, 18 the differences are 3, 5, 7, 9, which is not constant, so the sequence is not linear. The differences of the differences, the second differences, are all 2. A constant second difference means a quadratic sequence, and its nth term has an $n^{2}$ in it.

The method has two steps. First, halve the second difference to get the coefficient of $n^{2}$ : $2 \div 2 = 1$ , so the rule starts $n^{2}$ . Second, subtract $n^{2}$ from each term and look at what is left: 3 - 1 = 2, 6 - 4 = 2, 11 - 9 = 2, 18 - 16 = 2. The remainder is a constant 2, so $T_{n} = n^{2} + 2$ .

The syllabus example is 2, 5, 10, 17: the differences are 3, 5, 7 and the second difference is 2, so the coefficient of $n^{2}$ is 1 and the nth term is $n^{2} + 1$ . Check it: $n = 4$ gives $16 + 1 = 17$ .

If the second difference is 4 then the coefficient of $n^{2}$ is 2, and the sequence grows twice as fast. In 3, 10, 21, 36 the differences are 7, 11, 15 and the second difference is 4, so the rule starts $2 n^{2}$ ; subtracting $2 n^{2}$ leaves 1, 2, 3, 4, which is the linear sequence $n$ , so $T_{n} = 2 n^{2} + n$ .

3, 6, 11, 18: second difference 2, so $T_{n} = n^{2} + 2$
2, 5, 10, 17: second difference 2, so $T_{n} = n^{2} + 1$
3, 10, 21, 36: second difference 4, so $T_{n} = 2 n^{2} + n$
Half the second difference is the coefficient of $n^{2}$

A linear sequence has a constant first difference; a quadratic sequence has a constant second difference. Work out both rows of differences before deciding which one you have.

Cubic, exponential and combined sequences (Extended only)

When the third row of differences is constant, the sequence is cubic and its nth term has an $n^{3}$ in it. The cubes themselves, 1, 8, 27, 64, have a constant third difference of 6 and nth term $n^{3}$ . The sequence 2, 9, 28, 65 is the same shape shifted up by 1, so $T_{n} = n^{3} + 1$ .

An exponential sequence multiplies by the same number each time instead of adding. In 3, 6, 12, 24 the common ratio is 2 and the nth term is $T_{n} = 3 \times 2^{n - 1}$ . The index is $n - 1$ , not $n$ , because the first term already carries the 3 as its coefficient: $n = 1$ gives $3 \times 2^{0} = 3$ .

A simple combination of two sequences is built by adding or subtracting them term by term. The squares 1, 4, 9, 16 with 1 added give 2, 5, 10, 17. The powers of 2, which are 2, 4, 8, 16, with 1 added give 3, 5, 9, 17. If you recognise the sequence underneath, the nth term is nearly finished.

Deciding which family you have is most of the work: a constant first difference is linear, a constant second difference is quadratic, a constant third difference is cubic, and a constant ratio is exponential.

Family	Test on the differences	Form of the nth term	Example
Linear	first difference constant	$a n + b$	$4 n + 3$
Quadratic	second difference constant	$a n^{2} + b n + c$	$n^{2} + 2$
Cubic	third difference constant	$a n^{3} + b n^{2} + c n + d$	$n^{3} + 1$
Exponential	each term is the previous term times a fixed number	$a \times r^{n - 1}$	$3 \times 2^{n - 1}$

The ratio test is the one students forget. 2, 4, 8, 16 is not the sequence

2 n

; it is

2^{n}

, because each term is double the one before it.

Is a number in the sequence?

A question that asks whether a number is a term of a sequence is a solving question in disguise. Form an equation with the nth term equal to that number, and solve it for $n$ .

If the nth term is $3 n + 2$ , then 98 is a term only if $3 n + 2 = 98$ has a whole-number solution: $3 n = 96$ , so $n = 32$ , and yes, 98 is the 32nd term. For 100 the same equation gives $3 n = 98$ and $n = 32.67$ , which is not a whole number, so 100 is not a term of the sequence.

The condition is that $n$ must be a positive integer. A solution of $n = 32.67$ , or a negative one, proves the number is not in the sequence, and saying why is worth the mark: 'no, because $n$ is not a whole number'.

The same method works with a quadratic nth term. If $T_{n} = n^{2} + 1$ , then 50 is a term only if $n^{2} = 49$ , so $n = 7$ , and 50 is the 7th term. For 40 the equation gives $n^{2} = 39$ , and 39 is not a perfect square, so 40 is not a term.

$3 n + 2 = 98$ gives $n = 32$ , so 98 is in the sequence
$3 n + 2 = 100$ gives $n = 32.67$ , so 100 is not in the sequence
$n^{2} + 1 = 50$ gives $n = 7$ , so 50 is the 7th term
$n^{2} + 1 = 40$ gives $n^{2} = 39$ , so 40 is not in the sequence

Show the equation and the value of

n

. 'No, because

n

is not a whole number' scores; a bare 'no' does not.

Key terms you must be able to define

Sequence: A list of numbers in a definite order, such as 4, 7, 10, 13. Changing the order gives a different sequence.

Term: One number in a sequence. In 4, 7, 10, 13 the third term is 10.

Term-to-term rule: The rule for getting from one term to the next, such as 'add 3' or 'multiply by 2'. It continues the sequence one step at a time.

Position-to-term rule: The rule that gives any term from its position number, written as a formula in $n$ . It is another name for the nth term.

nth term: The general term of a sequence, written $T_{n}$ . Substituting a value of $n$ gives the term in that position.

Subscript notation: $T_{n}$ is the $n$ th term of sequence T, so $T_{1}$ is the first term and $T_{5}$ the fifth. A rule such as $T_{n + 1} = 2 T_{n} + 1$ links consecutive terms. Extended only.

Common difference: The constant gap between consecutive terms of a linear sequence: 4 in 7, 11, 15, 19. It is the coefficient of $n$ in the nth term.

Linear sequence: A sequence with a constant difference between consecutive terms, whose nth term has the form $a n + b$ . Also called an arithmetic sequence.

Quadratic sequence: A sequence with a constant second difference, whose nth term has the form $a n^{2} + b n + c$ . Its terms grow by increasing amounts.

Second difference: The difference between consecutive first differences. A constant second difference of 2 means the coefficient of $n^{2}$ is 1.

Cubic sequence: A sequence with a constant third difference, whose nth term contains an $n^{3}$ term. The cubes 1, 8, 27, 64 are the simplest example.

Exponential sequence: A sequence in which each term is the previous term multiplied by a fixed number, such as 3, 6, 12, 24. Its nth term has the form $a \times r^{n - 1}$ . Extended only.

Common ratio: The fixed multiplier in an exponential sequence: 2 in 3, 6, 12, 24, because each term is double the one before.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

Next two terms

Write down the next two terms of 2, 6, 12, 20. The differences are 4, 6, 8, and they grow by 2 each time, so the next difference is 10: 20 + 10 = 30, then 30 + 12 = 42.

Linear nth term

Find the nth term of 7, 11, 15, 19. The common difference is 4, so the rule starts $4 n$ . Substituting $n = 1$ gives $4 + b = 7$ , so $b = 3$ and $T_{n} = 4 n + 3$ . Check on the second term: $4 (2) + 3 = 11$ .

Decreasing linear sequence

Find the nth term of 20, 17, 14, 11. The common difference is -3, so the rule starts $- 3 n$ . At $n = 1$ , $- 3 + b = 20$ , so $b = 23$ and $T_{n} = 23 - 3 n$ . Check: $23 - 3 (2) = 17$ .

Quadratic nth term

Find the nth term of 3, 6, 11, 18. The first differences are 3, 5, 7, so the second difference is 2 and the coefficient of $n^{2}$ is 1. Subtracting $n^{2}$ from each term leaves 2, 2, 2, 2, so $T_{n} = n^{2} + 2$ . Check: $n = 4$ gives $16 + 2 = 18$ .

Is it a term?

The nth term of a sequence is $3 n + 2$ . Is 98 a term of the sequence? Solve $3 n + 2 = 98$ , so $3 n = 96$ and $n = 32$ . Since 32 is a positive whole number, 98 is the 32nd term of the sequence.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What is the difference between a term-to-term rule and a position-to-term rule?

A term-to-term rule gets from one term to the next, such as 'add 3'. A position-to-term rule, the nth term, gives any term straight from its position.

Q2Write down the next two terms of 2, 6, 12, 20.

30 and 42. The differences are 4, 6, 8, 10, 12, so the gaps grow by 2 each time.

Q3Find the nth term of 7, 11, 15, 19.

T_{n} = 4 n + 3

. The common difference 4 gives the 4, and

n = 1

needs 3 more to reach 7.

Q4Find the nth term of 20, 17, 14, 11.

T_{n} = 23 - 3 n

. The common difference is -3, and

23 - 3 (1) = 20

Q5What does a constant second difference tell you?

That the sequence is quadratic, so its nth term contains an

n^{2}

term. Halve the second difference to get the coefficient of

n^{2}

Q6Find the nth term of 3, 6, 11, 18.

T_{n} = n^{2} + 2

. First differences 3, 5, 7; second difference 2; subtracting

n^{2}

leaves 2 each time.

Q7What is the nth term of the sequence 2, 5, 10, 17?

T_{n} = n^{2} + 1

. The second difference is 2, and each term is one more than a square number.

Q8How do you decide whether 100 is a term of the sequence with nth term

3 n + 2

Solve

3 n + 2 = 100

. Here

n = 32.67

, which is not a whole number, so 100 is not a term.

Q9Is 98 a term of the sequence with nth term

3 n + 2

Yes.

3 n + 2 = 98

gives

n = 32

, a positive whole number, so 98 is the 32nd term.

Q10What test tells you a sequence is cubic?

The third row of differences is constant. The simplest cubic sequence is 1, 8, 27, 64, with nth term

n^{3}

Q11What is the nth term of 3, 6, 12, 24?

T_{n} = 3 \times 2^{n - 1}

. The common ratio is 2, and the index is

n - 1

so that

n = 1

gives the first term.

Q12What does

T_{5} = 23

mean?

The fifth term of sequence T is 23. The subscript is the position, not the value.

0 of 12 marked as known

Exam technique for this topic

Write the differences under the terms as your first line of working. That single row tells you whether the sequence is linear, quadratic or cubic, and it usually carries a method mark.
For a linear sequence the common difference is the coefficient of $n$ , and the constant is whatever makes $n = 1$ give the first term. Check with $n = 2$ before you write the answer down.
For a quadratic sequence, halve the second difference to get the coefficient of $n^{2}$ , then subtract $n^{2}$ (or $2 n^{2}$ ) from each term and find the rule for what is left. Guessing the constant loses the accuracy mark.
When a question asks whether a number is a term of a sequence, set the nth term equal to the number and solve. The answer has to be a positive whole number, and that reason belongs in your answer, not just in your head.
Extended only: an exponential sequence has a constant ratio, and its nth term is $a \times r^{n - 1}$ . Substitute $n = 1$ to check the index, because it must give the first term.

Why students lose marks here

A student wroteThe nth term of 5, 8, 11, 14 is

3 n

Why it loses marksThe common difference gives the 3, but

3 \times 1 = 3

, and the first term is 5. The constant has to be found by substituting

n = 1

, not ignored.

The answer that scores

T_{n} = 3 n + 2

, because

3 (1) + 2 = 5

and

3 (2) + 2 = 8

A student wroteThe nth term of 20, 17, 14, 11 is

3 n + 17

Why it loses marksThe sequence decreases, so the coefficient of

n

is negative. The common difference is -3, and using +3 makes the sequence increase instead.

The answer that scores

T_{n} = 23 - 3 n

, because

23 - 3 (1) = 20

and

23 - 3 (2) = 17

A student wroteFor 3, 6, 11, 18 the nth term is

n^{2} + 3

, because the first term is 3.

Why it loses marksThe second difference 2 does give

n^{2}

, but the constant is not the first term:

n^{2}

already contributes 1 when

n = 1

. Subtract

n^{2}

from each term and read off what is left.

The answer that scores

T_{n} = n^{2} + 2

, because 3 - 1 = 2, 6 - 4 = 2, 11 - 9 = 2 and 18 - 16 = 2.

A student wroteIs 100 a term of the sequence with nth term

3 n + 2

? Yes, because 3 x 33 = 99, which is nearly 100.

Why it loses marksA number is a term only if the equation has a whole-number solution.

3 n + 2 = 100

gives

n = 32.67

, and being close to a term is not the same as being one.

The answer that scores

3 n + 2 = 100

gives

3 n = 98

and

n = 32.7

to 3 significant figures, which is not a whole number, so 100 is not a term of the sequence.

Syllabus coverage

Term-to-term rules: what comes next

Position-to-term rules and the nth term

Finding the nth term of a linear sequence

Quadratic sequences: when the second difference is constant

Cubic, exponential and combined sequences (Extended only)

Is a number in the sequence?

Key terms you must be able to define

In context

Flashcards

Exam-style questions

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Exam technique for this topic

Why students lose marks here