Geometric Sequences (AI HL)
A geometric sequence multiplies by the same fixed amount, the common ratio, from one term to the next. This page covers the term formula, the sum of \(n\) terms, and the special case of a sum to infinity, with worked examples and the mistakes that lose the most marks. It's part of the broader Sequences & Series topic.
19 questions on this sub-topic.
The key formulas
Covered under IB syllabus reference SL1.3: geometric sequences and series, the \(n\)th-term and sum formulas, sigma notation, and applications such as disease spread, salary changes, and population growth. All three formulas below are in the formula booklet.
nth term
\(u_n = u_1 r^{n-1}\)
Use when you need one specific term - "find the 8th term", "find the value after 5 years of growth".
Sum of n terms
\(S_n = \dfrac{u_1(r^n-1)}{r-1},\ r\neq1\)
Use when you need a total across several terms rather than a single value.
Sum to infinity
\(S_\infty = \dfrac{u_1}{1-r},\ |r|<1\)
Only valid when the common ratio has magnitude less than 1, so successive terms shrink towards zero.
Need the full syllabus wording and formula-booklet reference table? See Sequences & Series.
Worked examples
A geometric sequence has first term \(3\) and common ratio \(2\).
(a) Find the 8th term.
(b) Find the sum of the first 8 terms.
Worked solution
(a) \(u_8 = 3(2)^{7}\) M1
\(= 384.\) A1
(b) \(S_8 = 3\dfrac{2^{8}-1}{2-1}\) M1
\(= 765.\) A1
A geometric series has first term \(20\) and common ratio \(0.4\).
(a) Explain why the sum to infinity exists.
(b) Find the sum to infinity.
Worked solution
(a) \(|r| = 0.4 < 1\), so the series converges. R1 A1
(b) \(S_\infty = \dfrac{a}{1-r} = \dfrac{20}{0.6} = \tfrac{100}{3}\) M1
\(\approx 33.3.\) A1
Plan A pays $1000 then +$200 each year; Plan B pays $1000 then +5% each year. Compare the total over 10 years.
(a)(i) State the total for Plan A over 10 years.
(a)(ii) State the total for Plan B over 10 years.
Worked solution
(a)(i) \(S_{10} = 5(2000 + 9\cdot200) = $19\,000.\) M1 A1
(a)(ii) \(S_{10} = \dfrac{1000(1.05^{10} - 1)}{0.05} \approx $12\,578.\) M1 A1
Plan A pays more. A1
Common mistakes
- Mixing up \(d\) and \(r\). Using the arithmetic formulas on a geometric sequence, or vice versa, is the single most common error - check whether consecutive terms have a constant difference or a constant ratio before picking a formula.
- Quoting a sum to infinity when \(|r| \geq 1\). The formula \(S_\infty = \dfrac{u_1}{1-r}\) only makes sense when the terms are shrinking. Always state and justify \(|r|<1\) before using it - IB mark schemes usually award a mark just for that check.
- Losing the negative sign in \(r\). When the common ratio is negative, forgetting to carry the sign through \(r^{n-1}\) flips the sign of every other term. Substitute carefully, especially on a non-calculator paper.
Ready to practise properly?
19 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.
When does a geometric series have a sum to infinity?
Only when \(|r| < 1\). In that case \(S_\infty = \dfrac{u_1}{1-r}\); if \(|r| \geq 1\) the terms don't shrink and there is no finite sum.