Home › IB Maths AA SL › Questions by topic › Sequences and Series

IB Maths AA SL · Unit 1: Number and Algebra

IB Maths AA SL Sequences and Series Questions

Exam-style IB Maths AA SL sequences and series questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.

Practise Sequences and Series questions → AA SL formula booklet

What you need to know

SL AA sequences questions are more algebraic than SL AI — expect proofs of formulas and sums to infinity for |r| < 1. Practise the derivation, not just the formula. Arithmetic and geometric sequences overview →

What's examined in AA SL sequences and series

The question bank covers these sequences and series question types (number of questions in brackets):

Key formulas

The n-th term of an arithmetic sequence
\(u_n = u_1 + (n-1)d\)
Sum of n terms of an arithmetic sequence
\(S_n = \tfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr) = \tfrac{n}{2}(u_1 + u_n)\)
The n-th term of a geometric sequence
\(u_n = u_1 \cdot r^{\,n-1}\)
Sum of n terms of a geometric sequence
\(S_n = \frac{u_1(r^n - 1)}{r - 1},\ r \ne 1\)
Infinite geometric series
\(S_\infty = \dfrac{u_1}{1 - r},\ |r| < 1\)

In the same notation as the IB formula booklet. All AA SL formulas →

Sequences and 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

1. Find the common ratio $r$ by dividing consecutive terms: $r = \frac{u_3}{u_2} = \frac{55}{44} = 1.25$.

2. Find the first term $u_1$ by working backwards: $u_1 = \frac{u_2}{r} = \frac{44}{1.25} = 35.2$.

3. State the formula for the sum of a geometric sequence: $S_n = \frac{u_1(r^n - 1)}{r - 1}$.

4. Substitute the known values to find $S_5$: $S_5 = \frac{35.2(1.25^5 - 1)}{1.25 - 1}$.

5. Evaluate the denominator and exponent: $S_5 = \frac{35.2(3.05175... - 1)}{0.25} = \frac{35.2(2.05175...)}{0.25}$.

6. Calculate the final sum: $S_5 = 288.8875 \implies \mathbf{289}$ (to 3 s.f.).

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 $4\text{ km}$ 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 $220\text{ km}$.

Solution

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.

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

1. Use the sum to infinity formula to find the common ratio: $S_\infty = \frac{u_1}{1 - r} \implies 200 = \frac{4}{1 - r}$.

2. Solve for $r$: $1 - r = \frac{4}{200} = 0.02 \implies r = 0.98$.

3. Set up the inequality for $S_n > 163$: $\frac{4(1 - 0.98^n)}{1 - 0.98} > 163$.

4. Simplify the fraction: $\frac{4(1 - 0.98^n)}{0.02} > 163 \implies 200(1 - 0.98^n) > 163$.

5. Isolate the exponential term: $1 - 0.98^n > 0.815 \implies 0.98^n < 0.185$.

6. Solve using logarithms (remembering to flip the inequality because $\log(0.98)$ is negative): $n > \frac{\log(0.185)}{\log(0.98)} \implies n > 83.52...$ The least integer is $\mathbf{84}$.

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.

Try these IB Maths AA SL sequences and series questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are 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 \(27.5\).

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

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

  1. Find the first term, \(u_1\).

  2. Find \(u_6\).

  3. Find the value of the infinite sum of the sequence.

Attempt it and see the mark scheme →

Question 3 · hard · 8 marks · Paper 2

The eighth term, \(u_8\), of an arithmetic sequence is \(18\) and the common difference, \(d\), is \(2\).

  1. Find the first term and the value of \(u_{17}\).

  2. The first and \(17\text{th}\) 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.

Attempt it and see the mark scheme →

All 46 sequences and series questions with mark schemes →

FAQ

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

There are 46 exam-style sequences and series questions in the AA SL question bank (Paper 1: 32 · Paper 2: 14), graded 9 easy, 22 medium, 9 hard, 4 very hard, 2 starter. 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: 32 · Paper 2: 14. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA SL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

More AA SL Unit 1 topics

Sequences and Series in other IB Maths courses

← All IB Maths AA SL topics