Marie runs $4\text{ km}$ in her first week of training. She increases her distance by exactly $1.5\text{ km}$ each subsequent week. Find the week in which her cumulative total distance run reaches exactly $220\text{ km}$.

1. Identify the sequence parameters: $u_1 = 4$, $d = 1.5$, and we need to find $n$ when $S_n = 220$. 2. State the arithmetic sum formula: $S_n = \frac{n}{2}(2u_1 + (n-1)d)$. 3. Substitute the parameters into the equation: $220 = \frac{n}{2}(2(4) + 1.5(n-1))$. 4. Expand and simplify the equation: $440 = n(8 + 1.5n - 1.5) \implies 440 = n(1.5n + 6.5)$. 5. Rearrange to form a standard quadratic equation: $1.5n^2 + 6.5n - 440 = 0$. 6. Solve using your GDC's equation solver. The roots are $n = 15.11...$ and $n = -19.4...$ Since she reaches the target *during* the $16^{\text{th}}$ week, the answer is Week 16 .

In a geometric sequence, the third term is $160$ and the common ratio is $\frac{1}{4}$ . Find the first term, $u_1$ . Find $u_6$ . Find the value of the infinite sum of the sequence.

$u_3 = u_1 r^2 \implies 160 = u_1 \left(\frac{1}{4}\right)^2$ . (M1) $160 = u_1 \left(\frac{1}{16}\right) \implies u_1 = 2560$ . A1 $u_6 = 2560\left(\frac{1}{4}\right)^5$ . (M1) $u_6 = 2.5$ . A1 $S_\infty = \frac{2560}{1 - 0.25} = \frac{2560}{0.75}$ . (M1) $S_\infty = \frac{10240}{3} \approx 3413.33$ . A1

IB Maths AA SL · Unit 1: Number and Algebra

Sequences & Series (IB Maths AA SL): notes, exam questions and worked examples

Q: Are the sequence and series formulas in the IB formula booklet?

Yes. The AA SL booklet gives the nth term and sum of n terms for arithmetic and geometric sequences, and the sum to infinity. What it does not tell you is which one a question needs, so practise spotting term questions versus sum questions.

Q: When does a geometric series have a sum to infinity?

Only when the common ratio r is strictly between −1 and 1. Then the sum to infinity is u1 / (1 − r).

Q: How many IB Maths AA SL sequences and series questions are there?

There are 64 exam-style sequences and series questions in the AA SL question bank (Paper 1: 45 · Paper 2: 19), graded 15 easy, 30 medium, 13 hard, 6 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is sequences and series on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 45 · Paper 2: 19. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Updated October 2026 · By Pete Bromfield

An arithmetic sequence adds the same difference $d$ each time; a geometric sequence multiplies by the same ratio $r$ . AA SL questions ask for a term ( $u_{n}$ ), a sum ( $S_{n}$ ) or, for a geometric series with $| r | < 1$ , the sum to infinity. Decide first whether the question wants a term or a sum.

64 questions
Paper 1: 45
Paper 2: 19
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Sequences & Series notes

Arithmetic sequences and series

$u_{n} = u_{1} + (n - 1) d$ and $S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})$ . With $u_{1} = 5$ and $d = 3$ : $u_{20} = 62$ and $S_{20} = 670$ .

Two unknowns ( $u_{1}$ and $d$ ) need two facts: write each as an equation and solve them simultaneously.

Geometric sequences and series

$u_{n} = u_{1} r^{n - 1}$ and $S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1}$ for $r \neq 1$ . With $u_{1} = 3$ and $r = 2$ : $S_{6} = 189$ . Find $r$ by dividing one term by the term before it.

Sum to infinity

A geometric series converges only when $| r | < 1$ ; then $S_{\infty} = \frac{u_{1}}{1 - r}$ . For $12 + 6 + 3 + \dots$ , $r = \frac{1}{2}$ and $S_{\infty} = 24$ . If $r$ contains $x$ , the condition $| r | < 1$ gives the values of $x$ for which the sum exists.

Sigma notation

$\sum_{k = 1}^{n}$ means "add the terms for $k = 1, 2, \dots, n$ ". Count the terms carefully: $\sum_{k = 3}^{10}$ has $8$ terms, not 7. Write out the first two or three terms to see whether the series is arithmetic or geometric.

Finding n and applications

When the unknown is $n$ , set up the inequality and solve it with logs or a table on the GDC. $n$ is a whole number, so round to the value that actually meets the condition ("the first term greater than 1000" means round up). In context questions (savings, seats in rows, a bouncing ball), say which term or sum you are finding before you calculate.

Key formulas

Arithmetic sequence In the booklet: $u_{n} = u_{1} + (n - 1) d$
Arithmetic series In the booklet: $S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})$
Geometric sequence In the booklet: $u_{n} = u_{1} r^{n - 1}$
Geometric series, $r \neq 1$ In the booklet: $S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}$
Sum to infinity, $| r | < 1$ In the booklet: $S_{\infty} = \frac{u_{1}}{1 - r}$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Arithmetic sequence · Geometric sequence · Sum of a geometric series · Sum of an arithmetic series · Sum to infinity of a geometric series

Sequences & Series worked examples

Worked example 1: Finding the sum of a geometric sequence · easy

The second term, $u_{2}$ , of a geometric sequence is $44$ and the third term, $u_{3}$ , is $55$ . Find $S_{5}$ , the exact sum of the first $5$ terms of the sequence.

Solution

6 steps · 1 to fill in

Examiner tip: Check your formula booklet carefully; students under pressure frequently mix up the arithmetic and geometric sum formulas, completely invalidating their working.

Worked example 2: Contextual arithmetic sequences · medium

Marie runs $4 km$ in her first week of training. She increases her distance by exactly $1.5 km$ each subsequent week. Find the week in which her cumulative total distance run reaches exactly $220 km$ .

Solution

6 steps

Examiner tip: Look for keywords like "cumulative" or "total". If a question asks for the total distance run over

n

weeks, use

S_{n}

. If it asks for the distance run *in* the

n

th week, use

u_{n}

Worked example 3: Infinite series and inequalities · hard

The first term of an infinite geometric sequence is $4$ and the sum to infinity is $200$ . Find the least integer value of $n$ for which the sum of the first $n$ terms strictly exceeds $163$ .

Solution

6 steps · 1 to fill in

Examiner tip: When isolating

- r^{n}

in an inequality, dividing or multiplying by a negative number flips the inequality symbol. Taking logarithms and dividing by

\log (0.98)

flips it again! Keep careful track of your signs.

Sequences & Series exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 3 marks · Paper 1

The sum of the first $16$ terms of an arithmetic sequence is $920$ . Find the common difference, $d$ , of the sequence if the first term is $27.5$ .

Check your answer in practice →

Question 2 · medium · 6 marks · Paper 1

In a geometric sequence, the third term is $160$ and the common ratio is $\frac{1}{4}$ .

Find the first term, $u_{1}$ .
Find $u_{6}$ .
Find the value of the infinite sum of the sequence.

Full worked solution Free, no sign-in

$u_{3} = u_{1} r^{2} ⟹ 160 = u_{1} {(\frac{1}{4})}^{2}$ . (M1)
$160 = u_{1} (\frac{1}{16}) ⟹ u_{1} = 2560$ . A1
$u_{6} = 2560 {(\frac{1}{4})}^{5}$ . (M1)
$u_{6} = 2.5$ . A1
$S_{\infty} = \frac{2560}{1 - 0.25} = \frac{2560}{0.75}$ . (M1)
$S_{\infty} = \frac{10240}{3} \approx 3413.33$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 8 marks · Paper 2

The eighth term, $u_{8}$ , of an arithmetic sequence is $18$ and the common difference, $d$ , is $2$ .

Find the first term and the value of $u_{17}$ .
The first and $17 th$ terms of this arithmetic sequence are the third and fifth terms respectively of a geometric sequence. Find the possible values for the common ratio, $r$ , and the corresponding first terms of the geometric sequence.

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Where students lose marks on sequences & series

Examiner's noteSequences: term vs sum, counting n, and spotting the sequence

Where marks are lost: Candidates used Sₙ when a term (or a sum of two terms) was needed, used n = 5 where the 6th term was meant, did not see that a pattern formed an arithmetic series, missed that a bouncing distance counts up and down, and found the median of a finite geometric sequence at a non-existent "5.5th" term.

Examiners flagged this in 3 exam series (November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Sequences & Series on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Sequences and sums on your calculator

Sequences & Series FAQ

Are the sequence and series formulas in the IB formula booklet?

Yes. The AA SL booklet gives the nth term and sum of n terms for arithmetic and geometric sequences, and the sum to infinity. What it does not tell you is which one a question needs, so practise spotting term questions versus sum questions.

When does a geometric series have a sum to infinity?

Only when the common ratio r is strictly between −1 and 1. Then the sum to infinity is u1 / (1 − r).

How many IB Maths AA SL sequences and series questions are there?

There are 64 exam-style sequences and series questions in the AA SL question bank (Paper 1: 45 · Paper 2: 19), graded 15 easy, 30 medium, 13 hard, 6 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is sequences and series on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 45 · Paper 2: 19. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 1: Exponentials & Logarithms · Financial Math · Proof

Sequences and Series in other courses: AI SL

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Arithmetic sequences & series · Geometric sequences & series · Sigma notation.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL number and algebra knowledge organiser

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← All IB Maths AA SL topics

Sequences & Series (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Sequences & Series notes

Arithmetic sequences and series

Geometric sequences and series

Sum to infinity

Sigma notation

Finding n and applications

Key formulas

Sequences & Series worked examples

Worked example 1: Finding the sum of a geometric sequence · easy

Solution

Worked example 2: Contextual arithmetic sequences · medium

Solution

Worked example 3: Infinite series and inequalities · hard

Solution

Sequences & Series exam-style questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 8 marks · Paper 2

Where students lose marks on sequences & series

Sequences & Series on your calculator

Sequences & Series FAQ

Next steps

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