Geometric Sequences (AI SL)
A geometric sequence multiplies by the same fixed value, the common ratio, from one term to the next - it's the model behind compound growth, repeated percentage change, and bouncing-ball problems. This page covers the two formulas you need, worked examples, and the mistakes that cost the most marks. It's part of the broader Sequences & Series topic.
28 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL1.3. Both formulas come from the formula booklet, so the skill is recognising a geometric pattern and picking the right one, not memorising them.
nth term
\(u_n = u_1 r^{n-1}\)
Use when you need one specific term - "find the 4th term", "find the value after 8 years of growth".
Sum of n terms
\(S_n = \dfrac{u_1(r^n-1)}{r-1},\ r\ne1\)
Use when you need a total - "find the total saved over 8 years". Always check \(r \ne 1\) before applying it.
Need the full syllabus wording and formula-booklet reference table? See Sequences & Series.
Worked examples
A geometric sequence has \(u_1 = 6\) and \(r = 1.5.\) Find \(u_4\).
Worked solution
A constant ratio means geometric with \(u_1 = 6,\ r = 1.5.\) M1
: \(u_4 = 6(1.5)^{4-1} = 6(1.5)^3 = 6(3.375) = 20.25.\)Note the exponent is \(n-1\) A1 \(=3\), not \(4\) - the first term has no factor of \(r.\)
A person saves amounts forming a geometric sequence: $500 in year 1, then 10% more each year.
Find the total saved over 8 years.
Worked solution
Savings grow by \(10\%\) yearly, so geometric with \(u_1 = 500,\ r = 1.1,\ n\) M1 \(= 8.\) A1
\(S_8 = \frac{u_1(r^n - 1)}{r - 1} = \frac{500(1.1^8 - 1)}{1.1 - 1} = \frac{500(2.14359 - 1)}{0.1}\) M1 \(= \frac{500(1.14359)}{0.1}.\) A1
\(S_8 \approx $5717.94.\) A1
For each sequence state whether it is arithmetic or geometric and give the common difference or ratio.
(a) \(5, 9, 13, 17, \dots\)
(b) \(80, 40, 20, 10, \dots\)
Worked solution
(a) \(9-5 = 4,\ 13-9 = 4,\ 17-13 = 4\): constant difference \(\Rightarrow\) arithmetic, \(d\) A1
\(= 4.\) A1
(b) \(\tfrac{40}{80} = \tfrac12,\ \tfrac{20}{40} = \tfrac12,\ \tfrac{10}{20} = \tfrac12\): constant ratio \(\Rightarrow\) geometric, \(r = \tfrac12.\) A1
Common mistakes
- Mixing up \(d\) and \(r\). A constant difference means arithmetic (\(d\)); a constant ratio means geometric (\(r\)). Confusing the two applies the wrong pair of formulas entirely.
- Getting the exponent wrong in \(u_n\). The power on \(r\) is \((n-1)\), not \(n\) - the first term carries no factor of \(r\) at all.
- Rounding \(r^n\) too early. On growth problems like compound savings, rounding the power partway through the sum formula shifts the final total by a noticeable amount - keep full precision until the last step.
Ready to practise properly?
30 geometric-sequence questions, marked instantly like the real exam.
Quick answers
What is the formula for the nth term of a geometric sequence?
\(u_n = u_1 r^{n-1}\), where \(u_1\) is the first term and \(r\) is the common ratio.
What is the formula for the sum of a geometric series?
\(S_n = \dfrac{u_1(r^n-1)}{r-1}\), valid whenever \(r \ne 1\).
Where can I check whether my GDC is set up correctly for this topic?
See the GDC guidance on the full Sequences & Series page for list-and-sum key sequences on the TI-84, Casio, and TI-Nspire.