Geometric sequences — the rules with examples
What is a geometric sequence and how do you find its terms?
In a geometric sequence the ratio between any two consecutive terms is constant — it is called the common ratio, r. Instead of adding the same number each time (as in a linear sequence), you multiply by the same number each time. From that constant ratio you can build a short formula for any term of the sequence, and even a formula for the sum of the terms — sometimes for the sum to infinity. At GCSE you need to recognise and continue a geometric sequence; the sum formulae are A level.
Worked examples, step by step
The sequence is 3, 6, 12, 24. Show that it is geometric and find the nth term
- Check the ratio between consecutive terms: 6 ÷ 3 = 2, 12 ÷ 6 = 2, and 24 ÷ 12 = 2
- Every ratio equals 2, so this is a geometric sequence with r = 2
- Substitute into the nth-term formula: aₙ = a₁ · rⁿ⁻¹ = 3 · 2ⁿ⁻¹
- Check: the fourth term from the formula is a₄ = 3 · 2³ = 3 × 8 = 24 — it matches the given sequence
In a geometric sequence a₁ = 5 and r = 2. What is the sixth term?
- Substitute into the nth-term formula: aₙ = a₁ · rⁿ⁻¹
- For n = 6: a₆ = 5 · 2⁵
- Work out the power: 2⁵ = 32
- Answer: a₆ = 5 × 32 = 160
In a geometric sequence a₂ = 6 and a₅ = 48. Find r and a₁
- Between the second and the fifth term there are 3 "jumps" of multiplying by r, so a₅ ÷ a₂ = r³
- Substitute: 48 ÷ 6 = 8 = r³, so r = 2
- Go back one step from a₂: a₁ = a₂ ÷ r = 6 ÷ 2 = 3
- Check: the sequence 3, 6, 12, 24, 48 — and the fifth term is indeed 48
Work out the sum 8 + 4 + 2 + 1 + ⋯
- Identify r: 4 ÷ 8 = 0.5, and check the pattern continues: 2 ÷ 4 = 0.5
- Because r = 0.5 and its size is less than 1, the sum to infinity exists
- Substitute into the sum-to-infinity formula: S = a₁ ÷ (1 − r) = 8 ÷ (1 − 0.5)
- Answer: S = 8 ÷ 0.5 = 16
Now try it yourself
Frequently asked questions
What is the difference between a geometric sequence and a linear (arithmetic) sequence?
In a linear sequence you add the same constant difference at every step; in a geometric sequence you multiply by the same constant ratio r. 5, 8, 11 is linear (add 3); 5, 10, 20 is geometric (multiply by 2).
What happens when r is negative?
The signs of the terms alternate. For example with a₁ = 2 and r equal to −3, the terms are 2, −6, 18, −54 — each term comes from the previous one by multiplying by r, sign included.
When does a geometric sequence have a sum to infinity?
Only when the size of r is less than 1, that is r lies strictly between −1 and 1. Then each term is smaller than the one before, the terms "shrink" towards zero and the sum converges to a finite number S = a₁ ÷ (1 − r). If |r| is 1 or more, there is no finite sum.
How do you prove that a sequence is geometric?
Work out the ratio aₙ₊₁ ÷ aₙ from the nth term and show that the result is a constant that does not depend on n. That is a stronger argument than checking a few early terms, which can mislead.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: