Sum of arithmetic series formula: Sₙ = n/2 (2a + (n − 1)d)
The sum of arithmetic series formula is Sₙ = n/2 × (2a + (n − 1)d) = n/2 × (a + l).
What each letter means
- \(S_n\) the sum of the first n terms
- \(a\) the first term
- \(d\) the common difference
- \(l\) the last term
- \(n\) the number of terms
When to use it
To add up the first n terms of an arithmetic sequence. Use the second form when you know the last term.
Worked example
Find the sum of all the multiples of 7 between 100 and 300.
- The terms are \(105, 112, \dots, 294\), so \(n=\dfrac{294-105}{7}+1=28\)
- \(S_{28}=\dfrac{28}{2}(105+294)=14\times399\)
Answer: \(5586\)
Common mistake
Halving only one part. The whole bracket is multiplied by n/2.
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 sum of arithmetic series formula?
The sum of arithmetic series formula is Sₙ = n/2 × (2a + (n − 1)d) = n/2 × (a + l). Sₙ: the sum of the first n terms; a: the first term; d: the common difference; l: the last term; n: the number of terms.
Is the sum of an arithmetic 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.
What is the difference between a sequence and a series?
A sequence is a list of terms. A series is what you get when you add the terms, so Sₙ is the sum of the first n terms.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.