Geometric sequence formula: uₙ = u₁rⁿ⁻¹
The geometric sequence formula is uₙ = u₁ rⁿ⁻¹.
What each letter means
- \(u_n\) the nth term
- \(u_1\) the first term
- \(r\) the common ratio: each term divided by the one before
- \(n\) the position of the term
When to use it
When each term is the previous one multiplied by the same number: compound growth, depreciation, bouncing balls, populations.
Worked example
A geometric sequence has first term 5 and common ratio 3. Find the 8th term.
- \(u_8=5\times3^{8-1}=5\times3^7=5\times2187\)
Answer: \(u_8=10\,935\)
Common mistake
Using rⁿ instead of rⁿ⁻¹. The first term has not been multiplied by r at all, so the 8th term has r⁷.
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 Geometric sequences (AI SL)
- Print AA SL one-page formula sheet
Questions
What is the geometric sequence formula?
The geometric sequence formula is uₙ = u₁ rⁿ⁻¹. uₙ: the nth term; u₁: the first term; r: the common ratio: each term divided by the one before; n: the position of the term.
Is the geometric 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 ratio?
Divide any term by the one before it: r = u₂ ÷ u₁. If you know two terms that are not next to each other, divide them to get a power of r and take the root.
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.