1.3 Geometric sequences and series questions for IB Maths AA

60 questions on Geometric sequences and series, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 1 Number and algebra
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	21	5	26
Paper 2	31	3	34
Every paper	52	8	60

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00685Paper 1 · 5 marks

In a geometric sequence, the second term is $12$ and the fifth term is $- \frac{3}{2}$ .

(a)
Find the common ratio.
3 marks
(b)
Find the sum to infinity of the sequence.
2 marks

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Question 2 · AA-00426Paper 2 · 15 marks

Asha and Bruno play a game in which they take turns to throw a dart at a small target, with Asha throwing first. The first player to hit the target wins and the game stops.

Each time Asha throws, the probability that she hits the target is 0.3. Each time Bruno throws, the probability that he hits the target is 0.4. All throws are independent.

(a)
Find the probability that Bruno wins with his first throw.
2 marks
(b)
Show that the probability that Asha wins with her $n$ th throw is $0.3 \times {0.42}^{n - 1}$ .
3 marks
(c)
Find the probability that Asha wins the game.
3 marks
(d)
Given that the game ends within the first three throws, find the probability that Asha wins.
4 marks
(e)
Find the least number of throws, $n$ , such that the probability that the game lasts for more than $n$ throws is less than 0.01.
3 marks

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Question 3 · AA-00686Paper 1 · 5 marks

Consider the recurring decimal $0.2454545 \dots$ , in which the digits $45$ repeat indefinitely.

(a)
Write $0.0454545 \dots$ as the sum of an infinite geometric series, stating its first term and its common ratio.
2 marks
(b)
Hence, write $0.2454545 \dots$ as a fraction in its simplest form.
3 marks

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Question 4 · AA-00687Paper 2 · 6 marks

The sum to infinity of a geometric series is $250$ . The sum of the first four terms of the series is $140$ .

(a)
Show that $r^{4} = 0.44$ , where $r$ is the common ratio.
2 marks
(b)
Find the two possible values of the first term of the series.
4 marks

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Question 5 · AA-00688Paper 1 · 6 marks

Consider the infinite geometric series

\sum_{k = 0}^{\infty} {(\frac{2 x - 3}{5})}^{k}, x \in R .

(a)
Find the values of $x$ for which the series converges.
3 marks
(b)
The sum to infinity of the series is $\frac{5}{4}$ . Find the value of $x$ .
3 marks

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Question 6 · AA-00690Paper 2 · 6 marks

The bob of a pendulum is released and swings freely. On its first swing, the bob travels along an arc of length $40$ cm. On each subsequent swing, the length of the arc is $92 %$ of the length of the arc on the previous swing.

(a)
Find the length of the arc on the sixth swing.
2 marks
(b)
Assuming that the bob swings indefinitely, find the total distance that the bob travels.
2 marks
(c)
Find the total distance that the bob travels from the end of its tenth swing onwards.
2 marks

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