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Quadratic Sequences (nth term)

HigherHigher tier onlyAQAEdexcelOCR

Quadratic Sequences (nth term) is a key algebra topic at GCSE Maths. This Higher worksheet gives you exam-style questions on finding the nth term of a linear sequence, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. A quadratic sequence has a constant second difference.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

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Topic overview

A quadratic sequence has a second difference that is constant rather than a first difference. Its nth term contains an \(n^2\) term, which is what produces the accelerating pattern.

The method has three stages. Halve the second difference to find the coefficient of \(n^2\), subtract that part from the sequence, then find the nth term of whatever linear sequence remains.

The subtraction step is where care is needed. Write the \(n^2\) values underneath the original sequence and subtract term by term, then treat the resulting row as an ordinary linear sequence. Combining the two parts gives the full formula.

Revision notes

Finding the n squared coefficient

Work out the first differences, then the differences of those. Halve the constant second difference.

For \(3, 8, 15, 24\): first differences \(5, 7, 9\); second difference \(2\). Half of \(2\) is \(1\), so the formula contains \(n^2\).

Subtracting the quadratic part

Write \(n^2\) for each position and subtract from the sequence.

Here \(n^2\) gives \(1, 4, 9, 16\). Subtracting from \(3, 8, 15, 24\) leaves \(2, 4, 6, 8\), which is linear.

Completing the formula

Find the nth term of the remaining linear sequence and add it to the quadratic part.

The leftover \(2, 4, 6, 8\) has nth term \(2n\), so the full formula is \(n^2 + 2n\). Check with \(n = 3\): \(9 + 6 = 15\), which matches.

Key points

  • A quadratic sequence has a constant second difference.
  • Halve the second difference for the \(n^2\) coefficient.
  • Subtract the quadratic part from the sequence.
  • The remainder is a linear sequence.
  • Combine the two parts for the full nth term.
  • Always check the formula against a given term.

Worked examples

Example 1

Find the nth term of \(4, 7, 12, 19, \ldots\)

Working

\[\text{2nd difference} = 2 \text{, so coefficient is } 1\]halve the second difference
\[4 - 1, 7 - 4, 12 - 9, 19 - 16 = 3, 3, 3, 3\]subtract n squared from each term
\[n^2 + 3\]the remainder is constant, so add 3

Example 2

Find the nth term of \(2, 8, 18, 32, \ldots\)

Working

\[\text{2nd difference} = 4 \text{, so coefficient is } 2\]halve the second difference
\[2n^2 \text{ gives } 2, 8, 18, 32\]write out the quadratic part
\[2n^2\]the remainder is zero, so this is the full formula

Example 3

Find the nth term of \(5, 12, 23, 38, \ldots\)

Working

\[\text{2nd difference} = 4 \text{, so coefficient is } 2\]halve the second difference
\[5 - 2, 12 - 8, 23 - 18, 38 - 32 = 3, 4, 5, 6\]subtract 2n squared from each term
\[2n^2 + n + 2\]the remainder has nth term n + 2

Common mistakes

  • Halving the first difference instead of the second.

    The n² coefficient comes from the second difference, which is constant in a quadratic sequence.

  • Forgetting to subtract before finding the linear part.

    The leftover sequence only appears once the quadratic part has been removed.

  • Stopping after finding the n² term.

    The formula usually has linear and constant parts too.

  • Not checking the finished formula.

    Substituting n = 1 and n = 3 catches most errors immediately.

Exam tips

  • Set the working out in rows: sequence, n², and the remainder.
  • Check the second difference really is constant before using this method.
  • Verify the completed formula against at least two given terms.
  • Remember the coefficient is half the second difference, not the difference itself.

Key terms

Quadratic sequence
A sequence whose nth term contains an \(n^2\) term.
Second difference
The difference between consecutive first differences.
First difference
The difference between consecutive terms.
Coefficient
The number multiplying a term such as \(n^2\).

Related topics

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