Sum of arithmetic series formula: Sₙ = n/2 (2u₁ + (n − 1)d)
The sum of arithmetic series formula is Sₙ = n/2 × (2u₁ + (n − 1)d) = n/2 × (u₁ + uₙ).
What each letter means
- \(S_n\) the sum of the first n terms
- \(u_1\) the first term
- \(d\) the common difference
- \(u_n\) the last term you add
- \(n\) how many terms you add
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 the first 30 terms of \(5+9+13+\cdots\)
- \(u_1=5,\ d=4,\ n=30\)
- \(S_{30}=15\,(10+29\times4)=15\times126\)
Answer: \(S_{30}=1890\)
Common mistake
Halving only one part. The whole bracket is multiplied by n/2.
On your course
| Course | In the exam |
|---|---|
| AA SL | In the IB formula booklet |
| AI SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA SL sequences and series questions
- Practise AI SL sequences and series questions
- Revise Sequences and series (AA SL)
- Revise Arithmetic sequences (AI SL)
- Tool Sums of sequences on your GDC
- Print AA SL one-page formula sheet
Questions
What is the sum of arithmetic series formula?
The sum of arithmetic series formula is Sₙ = n/2 × (2u₁ + (n − 1)d) = n/2 × (u₁ + uₙ). Sₙ: the sum of the first n terms; u₁: the first term; d: the common difference; uₙ: the last term you add; n: how many terms you add.
Is the sum of an arithmetic series given in the exam?
AA SL: in the IB formula booklet. AI SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.