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Maths · Algebra

Linear sequences

A linear sequence goes up or down by the same amount every time. Its nth term is a formula that gives any term straight away - the 100th as easily as the 1st - and tells you whether a number is in the sequence at all.

  • 6 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Write down the next term: 3, 7, 11, 15, ...

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    19

  2. 2

    What is the difference between the terms of 20, 17, 14, ...?

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    \(-3\)

  3. 3

    Solve \(3n + 2 = 50\).

    Show answerHide answer

    \(n = 16\)

  4. 4

    Last lesson: work out \(2n + 1\) when \(n = 10\).

    Show answerHide answer

    \(21\)

  5. 5

    Last lesson: make \(n\) the subject of \(T = 4n - 1\).

    Show answerHide answer

    \(n = \dfrac{T + 1}{4}\)

Learning Objectives

  1. 1Continue a sequence using a term-to-term rule.
  2. 2Generate terms of a sequence from its nth term.
  3. 3Find the nth term of a linear (arithmetic) sequence.
  4. 4Decide whether a number is a term of a sequence.
  5. 5Solve problems with linear sequences and patterns.

Sequences

A sequence is a list of numbers that follows a rule.

  • Term

    Each number in a sequence. The 1st term, 2nd term and so on.

  • Term-to-term rule

    How to get from one term to the next: "add 3".

  • Position-to-term rule (nth term)

    A formula for any term from its position \(n\): the nth term \(3n + 2\) gives 5, 8, 11, ...

  • Linear (arithmetic) sequence

    One that goes up or down by the same amount - the common difference - every time.

Where the nth Term Comes From

The sequence 5, 8, 11, 14, ... goes up by 3, so compare it with the 3 times table.

  • 1

    3n: 3. Term: 5. Term − 3n: \(+2\)

  • 2

    3n: 6. Term: 8. Term − 3n: \(+2\)

  • 3

    3n: 9. Term: 11. Term − 3n: \(+2\)

  • 4

    3n: 12. Term: 14. Term − 3n: \(+2\)

  • \(n\)

    3n: \(3n\). Term: \(3n + 2\). Term − 3n: \(+2\)

Finding the nth Term

Works for every linear sequence.

  1. 1 Difference

    Find the common difference, \(d\).
  2. 2 Times table

    Write down \(dn\): the \(d\) times table.
  3. 3 Compare

    How do you get from \(dn\) to the sequence?
  4. 4 Adjust

    Add or subtract that number.
  5. 5 Check

    Put \(n = 1\) and \(n = 2\) into your answer.

Finding the nth Term

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

Show the solutionHide the solution
  1. 1 The common difference is \(+4\) so it starts \(4n\)
  2. 2 The 4 times table 4, 8, 12, 16
  3. 3 Compare with the sequence \(7 - 4 = 3\): each term is 3 more
  4. 4 Write the nth term \(4n + 3\)
  5. 5 Check \(n = 3\) \(4 \times 3 + 3 = 15\)

Answer\(4n + 3\)

A Decreasing Sequence

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

Show the solutionHide the solution
  1. 1 The common difference is \(-3\) so it starts \(-3n\)
  2. 2 The \(-3\) times table \(-3\), \(-6\), \(-9\), \(-12\)
  3. 3 Compare with the sequence \(20 - (-3) = 23\): each term is 23 more
  4. 4 Write the nth term \(-3n + 23\), usually written \(23 - 3n\)
  5. 5 Check \(n = 1\) \(23 - 3 = 20\)

Answer\(23 - 3n\)

Is It in the Sequence?

Is 100 a term of the sequence with nth term \(4n + 3\)?

Show the solutionHide the solution
  1. 1 Set the nth term equal to 100 \(4n + 3 = 100\)
  2. 2 Solve \(4n = 97\), so \(n = 24.25\)
  3. 3 \(n\) must be a whole number (a position) 24.25 is not a whole number

AnswerNo - 100 is not in the sequence (the 24th term is 99 and the 25th is 103).

Using the Pattern

Pattern \(n\) uses \(3n + 1\) matchsticks. (a) How many matchsticks does pattern 20 use? (b) What is the largest pattern that can be made with 100 matchsticks?

Show the solutionHide the solution
  1. 1 (a) Substitute \(n = 20\) \(3 \times 20 + 1 = 61\)
  2. 2 (b) The number of matchsticks must be 100 or fewer \(3n + 1 \le 100\)
  3. 3 Solve \(3n \le 99\), so \(n \le 33\)
  4. 4 Pattern 33 uses exactly \(3 \times 33 + 1 = 100\) it fits

Answer(a) 61 matchsticks (b) Pattern 33

Sequence Detectives

For each sequence, find the nth term and the 50th term. Then decide whether 150 is in it. (a) 6, 11, 16, 21, ... (b) 1, 4, 7, 10, ... (c) 40, 36, 32, 28, ... (d) \(-5\), \(-3\), \(-1\), 1, ...

1. Find the difference.

2. Write the nth term and check it.

3. Solve nth term = 150.

A good answer shows: (a) \(5n + 1\); 251; 150 is not in it (\(n = 29.8\)). (b) \(3n - 2\); 148; 150 is not in it (\(n = 50.67\)). (c) \(44 - 4n\); \(-156\); 150 is not in it (it is decreasing from 40). (d) \(2n - 7\); 93; 150 is not in it (\(n = 78.5\)).

Can I...?

  1. 1Continue a sequence.
  2. 2Describe a term-to-term rule.
  3. 3Work out terms from an nth term.
  4. 4Find the nth term of an increasing sequence.
  5. 5Find the nth term of a decreasing sequence.
  6. 6Decide whether a number is in a sequence.
  7. 7Find the nth term of a pattern.
  8. 8Use an nth term to solve a problem.

Summary & Exam Focus

  • A linear sequence has a common difference.
  • nth term = (difference) \(\times\ n\), then adjust with a number.
  • To test a number, set the nth term equal to it: a whole-number \(n\) means it is in the sequence.
  • For patterns, the difference is how many more items each new pattern needs.

Exam focus

Here are the first five terms of a sequence: 5, 9, 13, 17, 21. Find an expression for the nth term. (2 marks) (2 marks)

"\(n + 4\)" is the term-to-term rule written wrongly, not an nth term. The nth term of a sequence going up by 4 always starts \(4n\).

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Sequence
A list of numbers or patterns that follows a rule.
Term
One number in a sequence.
Term-to-term rule
The rule that gets you from one term to the next.
nth term
A formula that gives any term from its position, \(n\).
Common difference
The amount a linear sequence goes up or down by each time.
Linear (arithmetic) sequence
A sequence with a common difference.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    Here are the first five terms of a sequence: 5, 9, 13, 17, 21. Find an expression, in terms of \(n\), for the nth term of the sequence.

    Show answerHide answer

    Model answer

    \(4n + 1\)

    Mark scheme

    • \(4n + 1\) — B2
    • \(4n + k\), where \(k \ne 1\) — B1
  2. Question 2 Non-calculator 2 marks

    Is 99 a term of the sequence 5, 9, 13, 17, 21, ...? You must give a reason for your answer.

    Show answerHide answer

    Model answer

    \(4n + 1 = 99\) gives \(n = 24.5\), which is not a whole number, so 99 is not a term. (The terms go 97, 101.)

    Mark scheme

    • \(4n + 1 = 99\), or listing terms to 97 and 101 — M1
    • No, with a correct reason — C1
  3. Question 3 Non-calculator 2 marks

    Find an expression for the nth term of the sequence 30, 26, 22, 18, ...

    Show answerHide answer

    Model answer

    \(34 - 4n\)

    Mark scheme

    • \(34 - 4n\) — B2
    • \(-4n + k\), where \(k \ne 34\) — B1
  4. Question 4 Calculator 3 marks

    The diagram shows the first three patterns in a sequence made from matchsticks. Pattern 1 uses 4 matchsticks, pattern 2 uses 7 and pattern 3 uses 10. Jack has 150 matchsticks. What is the largest pattern number he can make?

    Matchstick patterns of 1, 2 and 3 squares in a row, using 4, 7 and 10 matchsticks.
    Show answerHide answer

    Model answer

    The nth pattern uses \(3n + 1\) matchsticks. \(3n + 1 \le 150\), so \(3n \le 149\) and \(n \le 49.67\). The largest pattern is pattern 49 (it uses 148 matchsticks).

    Mark scheme

    • \(3n + 1\), or the difference of 3 used — P1
    • \(3n + 1 = 150\) or \(3n + 1 \le 150\), or \(49.6\ldots\) seen — P1
    • 49 — A1
  5. Question 5 Calculator 3 marks

    Sequence A has nth term \(3n + 5\). Sequence B has nth term \(50 - 2n\). For which value of \(n\) are the nth terms of the two sequences equal, and what is that term?

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    Model answer

    \(3n + 5 = 50 - 2n\), so \(5n = 45\) and \(n = 9\). The term is \(3 \times 9 + 5 = 32\).

    Mark scheme

    • \(3n + 5 = 50 - 2n\) — M1
    • \(n = 9\) — A1
    • 32 — A1

Quick check

  1. What is the 10th term of the sequence with nth term \(3n - 1\)?

    1. A30
    2. B31
    3. C29
    4. D27
    Show answerHide answer

    C: 29

    \(3 \times 10 - 1 = 29\).

  2. What is the nth term of 2, 7, 12, 17, ...?

    1. A\(5n - 3\)
    2. B\(n + 5\)
    3. C\(5n + 2\)
    4. D\(2n + 5\)
    Show answerHide answer

    A: \(5n - 3\)

    The difference is 5, so \(5n\): 5, 10, 15. Each term is 3 less: \(5n - 3\).

  3. What is the nth term of 15, 13, 11, 9, ...?

    1. A\(2n + 13\)
    2. B\(17 - 2n\)
    3. C\(15 - 2n\)
    4. D\(n - 2\)
    Show answerHide answer

    B: \(17 - 2n\)

    The difference is \(-2\), so \(-2n\): \(-2\), \(-4\), \(-6\). Each term is 17 more: \(17 - 2n\).

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