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Geometric Sequences

FoundationHigherAQAEdexcelOCR

Practise geometric sequences with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through geometric sequences and progressions, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Each term is multiplied by a fixed common ratio.

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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.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A geometric sequence multiplies by a fixed number each time, rather than adding. That fixed multiplier is the common ratio, usually written \(r\).

You find the ratio by dividing any term by the one before it. For \(2, 6, 18, 54\), each term is three times the last, so \(r = 3\). Because the terms multiply, they grow much faster than a linear sequence.

The ratio can be a fraction, which makes the terms shrink, and it can be negative, which makes them alternate in sign. Both appear in exam questions, so always check by dividing rather than assuming the pattern from the first two terms alone.

Revision notes

Finding the common ratio

Divide any term by the previous one. The answer should be the same throughout.

For \(5, 10, 20, 40\): \(10 \div 5 = 2\) and \(20 \div 10 = 2\), so \(r = 2\). Check more than one pair to be sure the sequence really is geometric.

Fractional and negative ratios

A ratio between \(0\) and \(1\) makes the terms decrease, so \(80, 40, 20, 10\) has \(r = \frac{1}{2}\).

A negative ratio makes the signs alternate, so \(3, -6, 12, -24\) has \(r = -2\). Watch the signs carefully when dividing.

Continuing the sequence

Multiply the last term by the ratio to extend the sequence, or divide to work backwards.

For \(2, 6, 18, 54\) the next term is \(54 \times 3 = 162\). The term before the first would be \(2 \div 3\).

Key points

  • A geometric sequence multiplies by a fixed ratio.
  • Find the ratio by dividing a term by the previous one.
  • A ratio above 1 makes terms grow.
  • A fractional ratio makes terms shrink.
  • A negative ratio makes signs alternate.
  • Check the ratio with more than one pair of terms.

Worked examples

Example 1

Find the common ratio and the next term of \(3, 12, 48, \ldots\)

Working

\[12 \div 3 = 4\]divide a term by the previous one
\[48 \div 12 = 4\]confirm the ratio with a second pair
\[48 \times 4 = 192\]multiply the last term by the ratio

Example 2

Find the next term of \(162, 54, 18, \ldots\)

Working

\[54 \div 162 = \tfrac{1}{3}\]the ratio is a fraction, so terms decrease
\[18 \times \tfrac{1}{3} = 6\]multiply the last term by the ratio

Example 3

Find the common ratio of \(4, -8, 16, -32, \ldots\)

Working

\[-8 \div 4 = -2\]divide, keeping track of the signs
\[16 \div -8 = -2\]confirm with a second pair
\[r = -2\]the negative ratio makes the signs alternate

Common mistakes

  • Subtracting instead of dividing.

    Geometric sequences multiply, so the ratio comes from division, not from the difference.

  • Assuming the ratio from one pair only.

    Check at least two pairs, since 2, 6, 18 and 2, 6, 10 start the same way.

  • Losing the sign with a negative ratio.

    In 4, −8, 16 the ratio is −2, and forgetting the minus gives the wrong terms.

  • Confusing the ratio with the difference.

    In 5, 10, 20 the difference changes but the ratio is constantly 2.

Exam tips

  • Divide consecutive terms and check the ratio repeats.
  • Watch the signs when the terms alternate.
  • State whether the sequence grows or shrinks as a sense-check.
  • Multiply forwards and divide backwards to extend a sequence either way.

Key terms

Geometric sequence
A sequence where each term is multiplied by a fixed number.
Common ratio
The fixed multiplier between consecutive terms.
Alternating
Changing sign from one term to the next.
Consecutive
Following one after another.

Related topics

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