Short answer
A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous one by a fixed common ratio. Knowing how to sum these terms quickly — using a single formula — saves time in the exam and opens up a range of problem-solving questions at Higher tier.
At a glance
- Key stage
- GCSE
- Subject
- Algebra
- Type
- How-to guide
- For
- Students
- Read time
- 5 min
- Last updated
- 14 October 2026
Where this fits
- Key Stage 3Years 7–9
- GCSEYears 10–11This article
Method at a glance
- Identify a, r and n
- Substitute into the formula
- Evaluate rⁿ
- Complete the arithmetic
- Check by adding the first few terms manually
What is a geometric series?
A geometric sequence is a list of numbers with a constant multiplier between consecutive terms. For example: 3, 6, 12, 24, 48 — each term is multiplied by 2. The multiplier is called the common ratio r.
A geometric series is what you get when you add the terms of that sequence:
3 + 6 + 12 + 24 + 48 = 93
For short sequences you could add term by term, but the formula is faster, especially when n is large or the terms are not whole numbers.
What is the formula for the sum of a geometric series?
The sum of the first n terms of a geometric series is:
Sₙ = a(1 − rⁿ) ÷ (1 − r)
Where:
- a is the first term
- r is the common ratio
- n is the number of terms
This formula works for any value of r except r = 1 (which would give division by zero, but a series with r = 1 is just n copies of the same number, so Sₙ = na).
An equivalent form, sometimes more convenient when r > 1, is:
Sₙ = a(rⁿ − 1) ÷ (r − 1)
Both forms produce the same result; use whichever feels more natural.
How do you apply the formula step by step?
- Identify a, r and n. Write down the first term, find r by dividing any term by the one before it, and count how many terms you are summing.
- Substitute into the formula. Write out Sₙ = a(1 − rⁿ) ÷ (1 − r) with your values in place.
- Evaluate rⁿ. Calculate r raised to the power n. For large powers, use the xⁿ key on your calculator.
- Complete the arithmetic. Work out the numerator, then divide by the denominator. Keep fractions exact where possible to avoid rounding errors.
- Check by adding the first few terms manually. If n is small (say, 4 or 5 terms), verify your answer by adding the terms directly.
Worked example: applying the formula
Find the sum of the first 6 terms of the geometric series: 2 + 6 + 18 + 54 + …
Step 1 — Identify values:
a = 2, r = 6 ÷ 2 = 3, n = 6
Step 2 — Substitute:
S₆ = 2(1 − 3⁶) ÷ (1 − 3)
Step 3 — Evaluate 3⁶:
3⁶ = 729
Step 4 — Complete the arithmetic:
S₆ = 2(1 − 729) ÷ (1 − 3) = 2 × (−728) ÷ (−2) = (−1456) ÷ (−2) = 728
Verification: 2 + 6 + 18 + 54 + 162 + 486 = 728 ✓
Finite vs infinite geometric series
Whether a geometric series has a finite sum depends on the value of r:
| Feature | Finite geometric series | Infinite geometric series |
|---|---|---|
| Number of terms | Fixed value of n | Series continues forever |
| When it has a sum | Always | Only when −1 < r < 1 ( |
| Formula | Sₙ = a(1 − rⁿ) ÷ (1 − r) | S∞ = a ÷ (1 − r) |
| Example | 2 + 6 + 18 (r = 3, 3 terms) | 8 + 4 + 2 + 1 + … (r = ½) |
| Result example | S₃ = 26 | S∞ = 8 ÷ (1 − ½) = 16 |
| What happens to terms | Terms grow (if r > 1) | Terms shrink towards zero |
At GCSE Higher tier, questions about infinite geometric series ask you to state the condition |r| < 1 and use the formula S∞ = a ÷ (1 − r).
How do you find the sum to infinity?
If the terms get smaller and smaller (because |r| < 1), the series converges to a finite limit:
S∞ = a ÷ (1 − r), provided |r| < 1
Example: Find the sum to infinity of: 12 + 4 + 4/3 + …
a = 12, r = 4 ÷ 12 = 1/3
S∞ = 12 ÷ (1 − 1/3) = 12 ÷ (2/3) = 12 × (3/2) = 18
If the series had r = 2 instead, the terms grow without limit and no finite sum exists.
How is this topic examined at GCSE?
Typical GCSE exam questions ask you to:
- Identify whether a sequence is geometric and find r.
- Find the sum of a specified number of terms using the formula.
- Determine whether an infinite series converges and, if so, find its sum.
- Set up and solve an equation involving Sₙ to find n or a.
On some Higher papers, you may need to apply the sum formula within a problem context, such as finding the total distance travelled by a bouncing ball where each bounce is r times the previous one.
Frequently asked questions
Q: Does the geometric series formula appear on the GCSE formula sheet?
No — neither AQA nor Edexcel include the geometric series formula on the GCSE formula sheet. You must memorise Sₙ = a(1 − rⁿ) ÷ (1 − r). Write it out at the start of every practice question until it becomes automatic.
Q: What if r is negative?
The formula still works when r is negative. Be careful with signs: a series with r = −2 alternates between positive and negative terms. Evaluate (−2)ⁿ carefully, remembering that a negative number raised to an even power is positive and to an odd power is negative.
Q: How do I find the number of terms n if I know the sum?
Set your sum formula equal to the given value of Sₙ and solve for rⁿ. Then take logarithms (at A-level) or use trial and improvement (at GCSE) to find n. In some GCSE questions, n will be small enough to find by inspection.
Q: What is the difference between a series and a sequence?
A sequence is an ordered list of terms (e.g. 3, 6, 12, 24). A series is the sum of those terms (e.g. 3 + 6 + 12 + 24). In everyday speech the words are often used interchangeably, but in maths the distinction matters, especially when discussing convergence.
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Key terms
- geometric sequence
- common ratio r
- geometric series
- Substitute into the formula
- Evaluate rⁿ
- Complete the arithmetic
- Verification