RevisionPrep FAQ

IB Maths: Geometric Sequences & Series FAQ

Q: What's the formula for the nth term of a geometric sequence?

The nth term formula is $u_n = u_1 r^{n-1}$, where $u_1$ is the first term and $r$ is the common ratio. It's given on page 1 of the IB Mathematics AA and AI formula booklet, so you don't need to memorise it — you need to know when to use it.

Q: What is the sum to infinity formula and when can you use it?

The sum to infinity is $S_\infty = \frac{u_1}{1-r}$, and it only exists when $-1 < r < 1$. Outside that range the series diverges and has no finite sum — examiners often build a mark specifically for stating this condition, not just applying the formula.

Q: How do you find the common ratio in a geometric sequence?

Divide any term by the term immediately before it: $r = \frac{u_{n+1}}{u_n}$. If you're only given non-consecutive terms (like $u_1$ and $u_5$), set up $u_5 = u_1 r^4$ and solve for $r$ — remembering that even roots can give two possible values.

Q: How do you tell if a sequence is geometric or arithmetic in an exam question?

Check the pattern between consecutive terms. If you get the same result by dividing (not subtracting) each term by the previous one, it's geometric. Arithmetic sequences have a constant common difference $d$; geometric sequences have a constant common ratio $r$ — test both before committing to a formula.

Q: What are the most common mistakes students make with geometric series in IB exams?

The three recurring errors: using $S_\infty$ when $|r| \geq 1$, mixing up $r^{n-1}$ with $r^n$ in the term formula, and forgetting that $r$ can be negative when solved from an even power. All three cost easy marks that have nothing to do with understanding the topic.

Q: How are geometric sequences applied in real-world IB exam questions?

IB loves compound interest, population growth/decay, and depreciation questions built on geometric sequences — a car losing 15% of its value yearly is a classic $r = 0.85$ setup. These appear in Paper 2 with GDC use expected, and often link to logarithms when solving for $n$.

Answered by RevisionPrep's IB Educators

Geometric sequences show up every year in Paper 1 and Paper 2, usually worth 5-8 marks, and they trip students up in the same three places. Here's how to answer them properly, whether you're doing AA or AI, SL or HL — answered by RevisionPrep's IB Educators.

Concept & Formulas

How do you answer geometric sequences & series questions in IB Maths?

Identify the common ratio r first — divide any term by the one before it. Then decide what's being asked: a specific term uses $u_{n} = u_{1} r^{n - 1}$ , a finite sum uses $S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1}$ , and an infinite sum (only if $∣ r ∣ < 1$ ) uses $S_{\infty} = \frac{u _{1}}{1 - r}$ .

Worked example: A sequence has $u_{1} = 3$ and $u_{4} = 24$ . Find $r$ and $S_{6}$ .

$u_{4} = u_{1} r^{3} \Rightarrow 24 = 3 r^{3} \Rightarrow r^{3} = 8 \Rightarrow r = 2$
$S_{6} = \frac{3 ( 2 ^{6} - 1 )}{2 - 1} = 3 (63) = 189$

Quick tip: always check whether $r$ comes out negative or fractional — students often assume it must be a whole number and lose the plot when it isn't.

What's the formula for the nth term of a geometric sequence?

The nth term formula is $u_{n} = u_{1} r^{n - 1}$ , where $u_{1}$ is the first term and $r$ is the common ratio. It's given on page 1 of the IB Mathematics AA and AI formula booklet, so you don't need to memorise it — you need to know when to use it.

Common mistake: writing $u_{n} = u_{1} r^{n}$ instead of $r^{n - 1}$ . Check it against $u_{1}$ itself — when $n = 1$ , the exponent must be zero, giving $u_{1} r^{0} = u_{1}$ . If your formula doesn't satisfy that check, you've got the exponent wrong.

What is the sum to infinity formula and when can you use it?

The sum to infinity is $S_{\infty} = \frac{u _{1}}{1 - r}$ , and it only exists when $- 1 < r < 1$ . Outside that range the series diverges and has no finite sum — examiners often build a mark specifically for stating this condition, not just applying the formula.

Worked example: $u_{1} = 8$ , $r = \frac{1}{4}$ . Since $∣ r ∣ < 1$ , $S_{\infty} = \frac{8}{1 - 0.25} = \frac{8}{0.75} = 10.67$ (3 s.f.).

Quick tip: if a question asks you to 'show that the series converges,' you must state $∣ r ∣ < 1$ explicitly as a line of working — the answer alone doesn't earn that mark.

How do you find the common ratio in a geometric sequence?

Divide any term by the term immediately before it: $r = \frac{u _{n + 1}}{u _{n}}$ . If you're only given non-consecutive terms (like $u_{1}$ and $u_{5}$ ), set up $u_{5} = u_{1} r^{4}$ and solve for $r$ — remembering that even roots can give two possible values.

Common mistake: with $u_{3} = 20$ and $u_{5} = 80$ , students write $r^{2} = 4$ and stop at $r = 2$ , forgetting $r = - 2$ also satisfies it. Always check whether the context (e.g. positive population growth) rules one solution out.

Exam Technique & Common Mistakes

How do you tell if a sequence is geometric or arithmetic in an exam question?

Check the pattern between consecutive terms. If you get the same result by dividing (not subtracting) each term by the previous one, it's geometric. Arithmetic sequences have a constant common difference $d$ ; geometric sequences have a constant common ratio $r$ — test both before committing to a formula.

I've marked scripts where a student spots a pattern like 2, 6, 18, 54, assumes 'increasing fast' means geometric without checking — which happens to be right here, but the habit of testing $u_{2} - u_{1}$ against $u_{2} \div u_{1}$ before choosing a formula saves you on the trickier hybrid questions.

What are the most common mistakes students make with geometric series in IB exams?

The three recurring errors: using $S_{\infty}$ when $∣ r ∣ \geq 1$ , mixing up $r^{n - 1}$ with $r^{n}$ in the term formula, and forgetting that $r$ can be negative when solved from an even power. All three cost easy marks that have nothing to do with understanding the topic.

Quick checklist before submitting a geometric sequences answer:

Did I check $∣ r ∣ < 1$ before using sum to infinity?
Did I use $r^{n - 1}$ , not $r^{n}$ , in the nth term formula?
If I solved for $r$ from an even power, did I consider the negative root?
Does my answer make sense in context (a bank balance shouldn't be negative)?

How are geometric sequences applied in real-world IB exam questions?

IB loves compound interest, population growth/decay, and depreciation questions built on geometric sequences — a car losing 15% of its value yearly is a classic $r = 0.85$ setup. These appear in Paper 2 with GDC use expected, and often link to logarithms when solving for $n$ .

Worked example: A machine depreciates 12% annually from an initial value of USD 20,000. After how many years does it drop below USD 8,000?

$20000 (0.88)^{n} < 8000 \Rightarrow 0.8 8^{n} < 0.4 \Rightarrow n > \frac{l n ( 0.4 )}{l n ( 0.88 )} \approx 7.17$

So it takes 8 full years. This is exactly the style of question that mixes geometric sequences with logs on Paper 2.

Is geometric sequences and series harder at HL than SL?

The core formulas are identical at SL and HL — the difference is context. HL students meet geometric series inside harder problems: proof by induction on $S_{n}$ , complex number applications, and combined sequences/series questions worth more marks in Paper 3-style investigations for AA HL.

According to the IB Mathematics: Analysis and Approaches guide (first exams 2025), sequences and series sit under SL/HL Topic 1 (Number and Algebra), with HL adding proof techniques and more demanding problem-solving contexts rather than new formulas.

Comparisons & Related Topics

What's the difference between geometric and arithmetic sequences?

Arithmetic sequences add a constant difference $d$ each term (2, 5, 8, 11…); geometric sequences multiply by a constant ratio $r$ each term (2, 6, 18, 54…). Their sum formulas differ too — arithmetic sums grow linearly, geometric sums can grow explosively or converge to a finite limit.

Do I need geometric sequences for both Maths AA and Maths AI?

Yes — geometric sequences and series appear in the syllabus for both Mathematics: Analysis and Approaches and Mathematics: Applications and Interpretation, at SL and HL. AI tends to frame questions more around financial and real-world contexts; AA leans more into pure algebraic manipulation and proof.

How much of the IB Maths exam is geometric sequences worth?

There's no fixed percentage published by the IB, but geometric sequences typically contribute one dedicated question worth 5-8 marks per paper, plus smaller appearances woven into financial maths or induction proofs at HL. It's a reliable, learnable topic rather than a major mark-share item.

Study Strategy & Resources

How can I get full marks on geometric sequences questions?

Full marks come from method precision, not just the right final number: state formulas before substituting, show the $∣ r ∣ < 1$ check for sum to infinity, and round only at the last step. Examiners award method marks even when a final answer is wrong, so showing structured working matters more than speed.

Quick tip: practise past paper questions where the same series appears in both a 'find $u_{n}$ ' part and a 'find $S_{\infty}$ ' part later — these compound questions are where marks get lost through carried-forward rounding errors. On revisionprep.com, topical worksheets group these multi-part geometric series questions together so you can drill the full sequence-to-series pipeline in one sitting rather than piecemeal.

What resources help most for practising geometric sequences and series?

The most effective prep combines the IB formula booklet (so students stop memorising what's already given), timed past-paper questions, and topic-specific worksheets that isolate geometric sequences before mixing them with other Topic 1 content. Revision Notes that walk through worked examples step-by-step help most when a student is stuck on where to start, not just what formula to use.

Parents often ask why their child 'knows the formulas but still loses marks' — it's almost always presentation: skipping the convergence check, or not showing the substitution step examiners are trained to look for under the IB's mark scheme conventions.

Arithmetic vs Geometric Sequences

Feature	Arithmetic	Geometric
Pattern	Add constant d	Multiply by ratio r
nth term	$u_{n} = u_{1} + (n - 1) d$	$u_{n} = u_{1} r^{n - 1}$
Sum formula	$S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d)$	$S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1}$
Infinite sum	Never converges	Converges if $∣ r ∣ < 1$

For step-by-step worked examples, topical worksheets, and full mock papers covering geometric sequences and series, explore the Mathematics AA & AI resources on revisionprep.com.