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Sequences (nth term)

FoundationHigherAQAEdexcelOCR

Master sequences (nth term) for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers finding the nth term of a linear sequence and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. For a linear sequence, the nth term is (common difference)n + (zero term).

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Topic overview

The nth term is a formula that gives any term of a sequence directly from its position, without listing the terms before it. It is also called the position-to-term rule.

For a linear sequence the formula is always \(dn + b\), where \(d\) is the common difference. The difference tells you the coefficient of \(n\) immediately, so \(4, 7, 10, 13\) starts as \(3n\).

The constant is then found by comparing \(3n\) with the actual sequence. The multiples of \(3\) are \(3, 6, 9, 12\), and each term of the sequence is one more, so the nth term is \(3n + 1\). Checking with a term you were given confirms it.

Revision notes

Finding the coefficient of n

The common difference becomes the number in front of \(n\).

For \(4, 7, 10, 13\) the difference is \(3\), so the formula begins \(3n\). A decreasing sequence gives a negative coefficient, so \(20, 17, 14\) begins \(-3n\).

Finding the constant

Write out the multiples of the difference and compare them with the sequence, term by term. The gap between them is the constant.

For \(4, 7, 10, 13\): the multiples of \(3\) are \(3, 6, 9, 12\), each one less than the sequence, so add \(1\), giving \(3n + 1\).

Using the nth term

Substitute a position to find that term, or set the formula equal to a value to test membership.

The \(50\)th term of \(3n + 1\) is \(151\). To check whether \(100\) is in the sequence, solve \(3n + 1 = 100\), giving \(n = 33\), a whole number, so yes.

Key points

  • The nth term gives any term from its position.
  • For a linear sequence it has the form \(dn + b\).
  • The common difference is the coefficient of \(n\).
  • Compare with multiples of the difference to find the constant.
  • A decreasing sequence gives a negative coefficient.
  • A whole-number solution means the value is in the sequence.

Worked examples

Example 1

Find the nth term of \(6, 11, 16, 21, \ldots\)

Working

\[\text{difference} = 5\]the common difference gives the coefficient of n
\[5n \text{ gives } 5, 10, 15, 20\]write out the multiples of 5
\[5n + 1\]each term is one more than the multiple

Example 2

Find the nth term of \(19, 15, 11, 7, \ldots\)

Working

\[\text{difference} = -4\]the sequence decreases, so the coefficient is negative
\[-4n \text{ gives } -4, -8, -12, -16\]write out the multiples
\[-4n + 23\]each term is 23 more than the multiple

Example 3

Is \(83\) a term of the sequence \(4n + 3\)?

Working

\[4n + 3 = 83\]set the nth term equal to the value
\[4n = 80 \text{, so } n = 20\]solve for n
\[\text{Yes, it is the 20th term}\]n is a whole number, so 83 is in the sequence

Common mistakes

  • Using the first term as the constant.

    For 4, 7, 10 the formula is 3n + 1, not 3n + 4. Compare with the multiples of 3.

  • Forgetting the negative for a decreasing sequence.

    19, 15, 11 has difference −4, so the formula starts −4n.

  • Not checking the formula.

    Substitute n = 1 and n = 2 to confirm it reproduces the sequence.

  • Concluding a value is in the sequence when n is not a whole number.

    If solving gives n = 12.5, the value is not a term.

Exam tips

  • Write the multiples of the difference under the sequence to find the constant.
  • Always check your formula with two given terms.
  • Use a negative coefficient for a decreasing sequence.
  • For membership questions, solve and check whether n is a positive whole number.

Key terms

nth term
A formula giving any term from its position.
Position-to-term rule
Another name for the nth term formula.
Coefficient
The number multiplying n.
Linear sequence
A sequence with a constant difference.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.