Geometric series sum formula: Sₙ = a(1 − rⁿ)/(1 − r)
The geometric series sum formula is Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1.
What each letter means
- \(S_n\) the sum of the first n terms
- \(a\) the first term
- \(r\) the common ratio
- \(n\) the number of terms
When to use it
To add up the first n terms of a geometric sequence: savings with regular deposits, a bouncing ball, repeated percentage growth.
Worked example
A geometric series has first term 80 and common ratio 0.75. Find the sum of the first 8 terms, to 2 decimal places.
- \(S_8=\dfrac{80(1-0.75^8)}{1-0.75}=320(1-0.75^8)\)
Answer: \(S_8=287.96\)
Common mistake
Using rⁿ⁻¹ in the sum. That power belongs to the nth term; the sum uses rⁿ.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | In the Edexcel formulae booklet |
| Edexcel International A Level | In Pearson’s IAL formulae booklet |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
Questions
What is the geometric series sum formula?
The geometric series sum formula is Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1. Sₙ: the sum of the first n terms; a: the first term; r: the common ratio; n: the number of terms.
Is the sum of a geometric series given in the exam?
Edexcel A Level (9MA0): in the Edexcel formulae booklet. Edexcel International A Level: in Pearson’s IAL formulae booklet. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
Which version should I use?
Both give the same answer. The (rⁿ − 1)/(r − 1) form avoids negatives when r > 1; the (1 − rⁿ)/(1 − r) form is neater when r < 1.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.