Arithmetic and geometric sequences

Now that you can find an nth term, use it to test numbers, solve pattern problems and spot geometric sequences.

Practise this lesson free (7 questions)

Is it in the sequence?

Set the nth term equal to the number and solve.
Is 78 in 4n + 2? 4n + 2 = 78, so 4n = 76 and n = 19. Yes, it is the 19th term.
Is 100 in 4n + 2? 4n = 98, so n = 24.5. No: n must be a whole number.

Geometric sequences

Each term is the previous term multiplied by the same number, the common ratio.
3, 6, 12, 24, ...: ratio 2.
80, 40, 20, 10, ...: ratio 12 (each term halves).
Find the ratio by dividing a term by the one before it.

Patterns

Shape patterns often make arithmetic sequences.
Squares made of matchsticks: 4, 7, 10, ... (each new square adds 3 sticks).
nth term: 3n + 1, so pattern 15 uses 3 × 15 + 1 = 46 sticks.
Pattern number n1234
Matchsticks471013
Worked example

Pattern 1 has 5 tiles, pattern 2 has 8 and pattern 3 has 11. How many tiles are in pattern 20?

  1. Difference 3, so start with 3n
  2. 3n gives 3, 6, 9. Add 2: nth term 3n + 2
  3. 3 × 20 + 2

Answer: 62 tiles

Key idea

Arithmetic sequences add the same amount; geometric sequences multiply by the same ratio. To test a number, set the nth term equal to it: a whole-number n means it is in the sequence.

Key words
nth term
A rule that gives any term of a sequence from its position n. For 5, 8, 11, 14 the nth term is 3n + 2.
Term
One part of an expression or sequence. In 3x + 5, the terms are 3x and 5.
Ratio
A way to compare amounts, written with a colon, such as 3 : 2.

All KS3 Mathematics key terms and definitions →

Common mistakes
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Extra practice

2 more questions

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  1. Find the next term of 96, 48, 24, 12, ...

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    Answer: 6

    Each term is half the one before: 12 ÷ 2 = 6.

  2. What is the common ratio of 5, 20, 80, 320, ...?

    Show the answer

    Answer: 4

    20 ÷ 5 = 4.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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