Arithmetic sequence formula: uₙ = u₁ + (n − 1)d
The arithmetic sequence formula is uₙ = u₁ + (n − 1)d.
What each letter means
- \(u_n\) the nth term
- \(u_1\) the first term
- \(d\) the common difference (the same amount added each time)
- \(n\) the position of the term
When to use it
When a sequence goes up or down by the same amount each time, to find any term, or to find which term has a given value.
Worked examples
1. An arithmetic sequence has \(u_1=7\) and \(d=4\). Find \(u_{20}\).
- \(u_{20}=7+19\times4\)
Answer: \(u_{20}=83\)
2. Which term of the same sequence is 203?
- \(7+(n-1)4=203\)
- \(n-1=49\)
Answer: \(n=50\): the 50th term
Common mistake
Using n instead of n − 1. The first term has had no differences added yet.
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 Sequences on your GDC
- Print AA SL one-page formula sheet
Questions
What is the arithmetic sequence formula?
The arithmetic sequence formula is uₙ = u₁ + (n − 1)d. uₙ: the nth term; u₁: the first term; d: the common difference (the same amount added each time); n: the position of the term.
Is the arithmetic sequence (nth term) 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.
How do I find the common difference?
Subtract any term from the next one: d = u₂ − u₁. It is negative if the sequence goes down.
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.