🔮 Algebra Realm · Sequences

Geometric sequences

Geometric sequences focuses on how to identify sequences formed by multiplying by a constant ratio. In this lesson, focus on a sequence records values in order.

← Back to Region Back to Dashboard
🏅
Geometric sequences Realm Badge Earned through mastery, boss success and strong scores — not by clicking through locks.

Player progress

Level 1 · Apprentice

0 / 100 XP

0 day streak 0% topic learning 0 completed topics 0 badges 0 boss wins 10 open sections4 practice levels

Learn · open now

Understand Geometric sequences

Geometric sequences focuses on how to identify sequences formed by multiplying by a constant ratio. In this lesson, focus on a sequence records values in order.

A sequence records values in order. A term-to-term rule explains the move from one term to the next, while the positions keep the order visible. For geometric sequences, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Geometric sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the geometric sequences representation visible until the final line.

Picture the idea

Run terms through a sequence machine and compare a step-by-step growth view with a position-to-term table. Use the model to explain one change you notice while working on geometric sequences.

Check as you go

Use the rule to regenerate at least two terms already shown in the sequence. Write that check beside the final geometric sequences answer.

Key vocabulary

termpositiondifferenceterm-to-term rulenth termgeometricsequences

Rules and key facts

3, 6, 12, 24 continues with 48 by multiplying by 2.

  • List positions above the terms.
  • Inspect the change pattern or substitute the required position.
  • Apply the rule carefully to the requested term.
  • Check against known terms before extending the pattern. Record the check explicitly for geometric sequences.

Step-by-step method

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
  4. Check against known terms before extending the pattern. Record the check explicitly for geometric sequences.

What you need first

  • Recognise the vocabulary: term, position, difference.
  • Be able to explain the purpose of geometric sequences before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Tile patterns
  • Predicting repeated growth

Visual / interactive

See the idea, then move it around

Skip to Practice

Run terms through a sequence machine and compare a step-by-step growth view with a position-to-term table. Use the model to explain one change you notice while working on geometric sequences.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
Responsive · validated · topic linked

Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Geometric sequences — The sequence starts 3, 5, 7. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 2. Add 2 once more: 7 + 2 = 9. Final answer: 9. Check: substitute or compare with the original information to confirm the result fits the question.

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
Secure example

Use the normal method

Given information: Geometric sequences — The sequence starts 3, 17, 31. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 14. Add 14 once more: 31 + 14 = 45. Final answer: 45. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Distinguish multiplicative growth from additive growth.

Challenge example

Stretch the idea

Given information: Geometric sequences — The sequence starts 3, 16, 29. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 13. Add 13 once more: 29 + 13 = 42. Final answer: 42. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Geometric sequences — The sequence starts 3, 15, 27. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 12. Add 12 once more: 27 + 12 = 39. Final answer: 39. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: State multiply by, not add, in the rule.

Common mistakes

  • Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering geometric sequences questions.
  • Starting position counting at zero without being told.

How to check your answer

Use the rule to regenerate at least two terms already shown in the sequence. Write that check beside the final geometric sequences answer.

Extension challenge

Create a geometric sequences problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

Practice · always open

Question practice with hints and full solutions

Choose a difficulty, answer questions, ask for hints, see the method, retry, or generate a similar question. XP rewards accurate work and improved scores.

I’m Stuck

Year 8 practice studio

Practice Studio

Foundation, secure, challenge and exam-style questions are available immediately with instant feedback.

Progress 0/0 Score 0 Streak 0
I’m Stuck

Answer the questions, then check your score.

Score tracking: Answer questions to start tracking your score.

Games · always open

Geometric sequences challenge

Use sequence machine controls to solve three checked geometric sequences rounds. Solve at least two of three marked rounds and use feedback to correct any error.

Press Start Game to enter a topic-specific maths arena.

Boss challenge

Challenge Geometric sequences Arcane Boss

The boss is available when you feel ready. Boss victory badges and legendary status still require a strong pass.

Geometric sequences Arcane Boss

Timed mixed-difficulty battle. Practice first if you want, or jump in and learn from feedback.

12 mixed questions5:00 timer×2 XP multiplier
Timer readyAwaiting battle

Study cards and flashcards · always open

Study the essentials quickly

Study cards

Core idea

Geometric sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the geometric sequences representation visible until the final line.

Tap to mark reviewed
Key vocabulary

term · position · difference · term-to-term rule · nth term · geometric · sequences

Tap to mark reviewed
Rules

List positions above the terms. Inspect the change pattern or substitute the required position. Apply the rule carefully to the requested term. Check against known terms before extending the pattern. Record the check explicitly for geometric sequences.

Tap to mark reviewed
Formula / fact

3, 6, 12, 24 continues with 48 by multiplying by 2.

Tap to mark reviewed
Foundation example

Given information: Geometric sequences — The sequence starts 3, 5, 7. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 2. Add 2 once more: 7 + 2 = 9. Final answer: 9. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Geometric sequences — The sequence starts 3, 17, 31. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 14. Add 14 once more: 31 + 14 = 45. Final answer: 45. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Geometric sequences — The sequence starts 3, 16, 29. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 13. Add 13 once more: 29 + 13 = 42. Final answer: 42. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Geometric sequences — The sequence starts 3, 15, 27. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 12. Add 12 once more: 27 + 12 = 39. Final answer: 39. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Common mistake

Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering geometric sequences questions.

Tap to mark reviewed
Exam tip

For geometric sequences, show the key representation before the final calculation. Use this final check: Use the rule to regenerate at least two terms already shown in the sequence.

Tap to mark reviewed
Real-world use

Tile patterns, Predicting repeated growth

Tap to mark reviewed
Checklist

I can explain geometric sequences, use the method, check for mistakes, and answer an exam-style question.

Tap to mark reviewed

Flashcards

I’m Stuck

Help for Geometric sequences

Use this whenever a question feels confusing. Nothing here is locked.

Simple explanation

Geometric sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the geometric sequences representation visible until the final line.

Think of geometric sequences as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
  4. Check against known terms before extending the pattern. Record the check explicitly for geometric sequences.

Hint 1

Start by naming the given information and the exact result required for geometric sequences.

Hint 2

List positions above the terms.

Full worked solution

Given information: Geometric sequences — The sequence starts 3, 5, 7. Find the next term. Method choice: use the geometric sequences method and show each step with the stated values. Calculation or reasoning: The common difference is 2. Add 2 once more: 7 + 2 = 9. Final answer: 9. Check: substitute or compare with the original information to confirm the result fits the question.

Method: List positions above the terms. → Inspect the change pattern or substitute the required position. → Apply the rule carefully to the requested term. → Check against known terms before extending the pattern. Record the check explicitly for geometric sequences.

Common mistake warning

Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering geometric sequences questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain geometric sequences in my own words.
  • I can use these words accurately: term, position, difference.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Using a term-to-term rule as if it were an nth-term formula.
  • I can apply this check: Use the rule to regenerate at least two terms already shown in the sequence.
🥉 Bronze Badge Foundation level completed
🥈 Silver Badge Apprentice level completed
🥇 Gold Badge Skilled level completed
💎 Platinum Badge Mastery test passed
🐉 Legendary Badge Boss battle defeated