Geometric sequences

Objective. Recognise and continue sequences with a common ratio.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

A geometric sequence multiplies by the same common ratio r to get from one term to the next, rather than adding a fixed amount. Find r by dividing any term by the one before it, then multiply repeatedly to continue the pattern. For 3, 6, 12, 24, … the ratio is 6 ÷ 3 = 2, so each term doubles the previous one, giving the next term as 24 × 2 = 48. Always divide using two different pairs of consecutive terms to confirm the ratio is truly constant.

Worked example

Find the next term of the geometric sequence 2, 6, 18, 54, …

1 $Divide consecutive terms: 6 \div 2 = 3$
2 $Check with another pair: 18 \div 6 = 3, so the common ratio is 3$
3 $Multiply the last term by the ratio: 54 \times 3$
4 $The next term is 162$

Answer: 162

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Pupils sometimes treat a geometric sequence as arithmetic and subtract terms instead of dividing, concluding 3, 6, 12 increases by 3 then 6. Always test by division first to see if a constant ratio fits better than a constant difference.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard