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IB Math AA HL Questionbank - Sequences & Series
2023 Prediction Exams and November 2022 Past Paper Solutions available now!
IB MATH AA HL - QUESTIONBANK
Sequences & Series
Arithmetic/Geometric, Sigma Notation, Applications, Compound Interest…
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EASY
Question 1
[Maximum mark: 6]
Consider an arithmetic sequence 2, 6, 10, 14, …
(a) Find the common difference, d.
[2]
(b) Find the 10th term in the sequence.
[2]
(c) Find the sum of the first 10 terms in the sequence.
[2]
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Question 2
[Maximum mark: 6]
An arithmetic sequence has u1 = 40, u2 = 32, u3 = 24.
​
​
​
(a) Find the common difference, d.
[2]
(b) Find u8 .
[2]
​
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IB Math AA HL Questionbank - Sequences & Series
(c) Find S8 .
[2]
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Question 3
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
an
=
​
​
bn
1, 5, 10, 15, …
​
=
​
1 2 3 4
, , , ,…
2 3 4 5
​
​
​
cn
​
=
​
​
​
dn
​
1.5, 3, 4.5, 6, …
​
=
​
2, 1,
1 1
, ,…
2 4
​
​
​
(a) State which sequence is arithmetic and find the common difference of the sequence.
[2]
(b) State which sequence is geometric and find the common ratio of the sequence.
[2]
(c) For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction.
[2]
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Question 4
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
an
=
​
​
bn
​
​
​
=
1 1 1 1
, , , ,…
3 4 5 6
​
​
cn
​
​
2.5, 5, 7.5, 10, …
=
​
​
​
dn
​
3, 1,
​
=
1 1
, ,…
3 9
​
​
​
1, 3, 6, 10, …
(a) State which sequence is arithmetic and find the common difference of the sequence.
[2]
(b) State which sequence is geometric and find the common ratio of the sequence.
[2]
(c) For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction.
[2]
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IB Math AA HL Questionbank - Sequences & Series
EASY
Question 5
[Maximum mark: 6]
Consider the infinite geometric sequence 4480, −3360, 2520, −1890, …
(a) Find the common ratio, r.
[2]
(b) Find the 20th term.
[2]
(c) Find the exact sum of the infinite sequence.
[2]
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Question 6
[Maximum mark: 6]
The table shows the first four terms of three sequences: un , vn , and wn .
​
​
​
(a) State which sequence is
(i) arithmetic;
(ii) geometric.
[2]
(b) Find the sum of the first 50 terms of the arithmetic sequence.
[2]
(c) Find the exact value of the 13th term of the geometric sequence.
[2]
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IB Math AA HL Questionbank - Sequences & Series
EASY
Question 7
[Maximum mark: 7]
An arithmetic sequence is given by 3, 5, 7, …
(a) Write down the value of the common difference, d.
[1]
(b) Find
(i) u10 ;
​
(ii) S10 .
[4]
​
(c) Given that un = 253, find the value of n.
[2]
​
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Question 8
[Maximum mark: 6]
Consider the infinite geometric sequence 9000, −7200, 5760, −4608, ...
(a) Find the common ratio.
[2]
(b) Find the 25th term.
[2]
(c) Find the exact sum of the infinite sequence.
[2]
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Question 9
[Maximum mark: 6]
A tennis ball bounces on the ground n times. The heights of the bounces, h1 , h2 , h3 , … , hn , form a geometric sequence.
The height that the ball bounces the first time, h1 , is 80 cm, and the second time, h2 , is 60 cm.
​
​
​
​
​
​
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IB Math AA HL Questionbank - Sequences & Series
(a) Find the value of the common ratio for the sequence.
[2]
(b) Find the height that the ball bounces the tenth time, h10 .
[2]
(c) Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer
correct to 2 decimal places.
[2]
​
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Question 10
[Maximum mark: 6]
The third term, u3 , of an arithmetic sequence is 7. The common difference of
the sequence, d, is 3.
​
(a) Find u1 , the first term of the sequence.
[2]
(b) Find u60 , the 60th term of sequence.
[2]
​
The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.
(c) Calculate the sixth term of the geometric sequence.
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Question 11
[Maximum mark: 6]
The fifth term, u5 , of a geometric sequence is 125. The sixth term, u6 , is 156.25.
​
(a) Find the common ratio of the sequence.
[2]
(b) Find u1 , the first term of the sequence.
[2]
(c) Calculate the sum of the first 12 terms of the sequence.
[2]
​
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IB Math AA HL Questionbank - Sequences & Series
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Question 12
[Maximum mark: 6]
The fourth term, u4 , of a geometric sequence is 135. The fifth term, u5 , is 81.
​
​
(a) Find the common ratio of the sequence.
[2]
(b) Find u1 , the first term of the sequence.
[2]
(c) Calculate the sum of the first 20 terms of the sequence.
[2]
​
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Question 13
[Maximum mark: 6]
The fifth term, u5 , of an arithmetic sequence is 25. The eleventh term, u11 , of the same sequence is 49.
(a) Find d, the common difference of the sequence.
[2]
(b) Find u1 , the first term of the sequence.
[2]
(c) Find S100 , the sum of the first 100 terms of the sequence.
[2]
​
​
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Question 14
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[Maximum mark: 6]
Consider the following sequence of figures.
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IB Math AA HL Questionbank - Sequences & Series
Figure 1 contains 6 line segments.
(a) Given that Figure n contains 101 line segments, show that n = 20.
[3]
(b) Find the total number of line segments in the first 20 figures.
[3]
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Question 15
[Maximum mark: 5]
Consider an arithmetic sequence where u12 = S12 = 12. Find the value of the first term, u1 , and the value of the
common difference, d.
​
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Question 16
[Maximum mark: 6]
In an arithmetic sequence, u5 = 24, u13 = 80.
​
​
(a) Find the common difference.
[2]
(b) Find the first term.
[2]
(c) Find the sum of the first 20 terms in the sequence.
[2]
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IB Math AA HL Questionbank - Sequences & Series
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Question 17
[Maximum mark: 6]
The first three terms of a geometric sequence are u1 = 32, u2 = −16, u3 = 8.
​
​
​
(a) Find the value of the common ratio, r.
[2]
(b) Find u6 .
[2]
​
(c) Find S∞ .
[2]
​
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Question 18
[Maximum mark: 6]
In an arithmetic sequence, u4 = 12, u11 = −9.
​
​
(a) Find the common difference.
[2]
(b) Find the first term.
[2]
(c) Find the sum of the first 11 terms in the sequence.
[2]
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Question 19
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[Maximum mark: 5]
In an arithmetic sequence, the sum of the 2nd and 6th term is 32.
Given that the sum of the first six terms is 120, determine the first term and common difference of the sequence.
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IB Math AA HL Questionbank - Sequences & Series
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Question 20
[Maximum mark: 5]
An arithmetic sequence has first term 45 and common difference −1.5.
(a) Given that the k th term of the sequence is zero, find the value of k .
[2]
Let Sn denote the sum of the first n terms of the sequence.
​
(b) Find the maximum value of Sn .
[3]
​
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Question 21
[Maximum mark: 6]
An arithmetic sequence has first term −30 and common difference 5.
(a) Given that the k th term is the first positive term of the sequence, find the value of k .
[3]
Let Sn denote the sum of the first n terms of the sequence.
​
(b) Find the minimum value of Sn .
[3]
​
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Question 22
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[Maximum mark: 6]
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IB Math AA HL Questionbank - Sequences & Series
The Australian Koala Foundation estimates that there are about 45 000 koalas left in the wild in 2019. A year before, in
2018, the population of koalas was estimated as 50 000. Assuming the population of koalas continues to decrease by the
same percentage each year, find:
(a) the exact population of koalas in 2022;
[3]
(b) the number of years it will take for the koala population to reduce to half of its number in 2018.
[3]
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Question 23
[Maximum mark: 6]
Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is 16.1 km.
The first landmark placed on the road is 124.7 km from London, and the last landmark is near Edinburgh. The length of
the road from London to Edinburgh is 667.1 km.
(a) Find the distance between the fifth landmark and London.
[3]
(b) Determine how many landmarks there are along the road.
[3]
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Question 24
[Maximum mark: 6]
The first term of an arithmetic sequence is 24 and the common difference is 16.
(a) Find the value of the 62nd term of the sequence.
[2]
The first term of a geometric sequence is 8. The 4th term of the geometric sequence is equal to the 13th term of the
arithmetic sequence given above.
(b) Write down an equation using this information.
[2]
(c) Calculate the common ratio of the geometric sequence.
[2]
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IB Math AA HL Questionbank - Sequences & Series
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EASY
Question 25
[Maximum mark: 6]
On 1st of January 2021, Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a
loan of $P with a bank that offers a nominal annual interest rate of 2.6%, compounded monthly.
The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, α.
(a) Find the value of α, giving your answer to five significant figures.
[3]
The bank lets the size of Fiona's loan increase until it becomes triple the size of the original loan. Once this happens,
the bank demands that Fiona pays the entire amount back to close the loan.
(b) Find the year during which Fiona will need to pay back the loan.
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Question 26
[Maximum mark: 6]
The first three terms of an arithmetic sequence are u1 , 4u1 − 9, and 3u1 + 18.
​
​
​
(a) Show that u1 = 9.
[2]
(b) Prove that the sum of the first n terms of this arithmetic sequence is a square number.
[4]
​
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Question 27
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[Maximum mark: 6]
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IB Math AA HL Questionbank - Sequences & Series
On Gary's 50th birthday, he invests $P in an account that pays a nominal annual interest rate of 5 %, compounded
monthly.
The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, α.
(a) Find the value of α, giving your answer to four significant figures.
[3]
Gary makes no further deposits or withdrawals from the account.
(b) Find the age Gary will be when the amount of money in his account will be double the amount he invested.
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Question 28
[Maximum mark: 7]
In an arithmetic sequence, the third term is 41 and the ninth term is 23.
(a) Find the common difference.
[2]
(b) Find the first term.
[2]
(c) Find the smallest value of n such that Sn < 0.
[3]
​
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Question 29
[Maximum mark: 7]
The first three terms of a geometric sequence are u1 = 0.8, u2 = 2.4, u3 = 7.2.
​
​
​
(a) Find the value of the common ratio, r.
[2]
(b) Find the value of S8 .
[2]
​
(c) Find the least value of n such that Sn > 35 000.
[3]
​
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IB Math AA HL Questionbank - Sequences & Series
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Question 30
[Maximum mark: 7]
The first three terms of a geometric sequence are u1 = 0.4, u2 = 0.6, u3 = 0.9.
​
​
​
(a) Find the value of the common ratio, r.
[2]
(b) Find the sum of the first ten terms in the sequence.
[2]
(c) Find the greatest value of n such that Sn < 650.
[3]
​
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Question 31
[Maximum mark: 7]
In a geometric sequence, u2 = 6, u5 = 20.25.
​
​
(a) Find the common ratio, r.
[2]
(b) Find u1 .
[2]
​
(c) Find the greatest value of n such that un < 200.
[3]
​
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Question 32
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[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
A population of goats on an island starts at 232. The population is expected to increase by 15 % each year.
(a) Find the expected population size after:
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IB Math AA HL Questionbank - Sequences & Series
(i) 10 years;
(ii) 20 years.
(b) Find the number of years it will take for the population to reach 15 000.
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Question 33
[Maximum mark: 5]
Consider an arithmetic sequence with u1 = 5 and u6 = log3 32.
​
​
​
Find the common difference of the sequence, expressing your answer in the form log3 a, where a ∈ Q.
​
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Question 34
[Maximum mark: 6]
On 1st of January 2022, Grace invests $P in an account that pays a nominal annual interest rate of 6 %, compounded
quarterly.
The amount of money in Grace's account at the end of each year follows a geometric sequence with common ratio, α.
(a) Find the value of α, giving your answer to four significant figures.
[3]
Grace makes no further deposits or withdrawals from the account.
(b) Find the year in which the amount of money in Grace's account will become triple the amount she invested.
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Question 35
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IB Math AA HL Questionbank - Sequences & Series
[Maximum mark: 6]
l
Consider the sum S = ∑(2k − 3), where l is a positive integer greater than 4.
​
k=4
(a) Write down the first three terms of the series.
[2]
(b) Write down the number of terms in the series.
[1]
(c) Given that S = 725, find the value of l.
[3]
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Question 36
[Maximum mark: 6]
Let un = 5n − 1, for n ∈ Z+ .
​
(a)
(i) Using sigma notation, write down an expression for u1 + u2 + u3 + ⋯ + u10 .
​
​
​
​
(ii) Find the value of the sum from part (a) (i).
[4]
A geometric sequence is defined by vn = 5 × 2n−1 , for n ∈ Z+ .
​
6
[2]
(b) Find the value of the sum of the geometric series ∑ vk .
​
​
k=1
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Question 37
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[Maximum mark: 6]
Peter is playing on a swing during a school lunch break. The height of the first swing was 2 m and every subsequent
swing was 84 % of the previous one. Peter's friend, Ronald, gives him a push whenever the height falls below 1 m.
(a) Find the height of the third swing.
[2]
(b) Find the number of swings before Ronald gives Peter a push.
[2]
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IB Math AA HL Questionbank - Sequences & Series
(c) Calculate the total height of swings if Peter is left to swing until coming
to rest.
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Question 38
[Maximum mark: 6]
Sarah walks to school each morning. During the first minute, she travels 130 metres. In each subsequent minute, she
travels 5 metres less than the distance she travelled during the previous minute. The distance from her home to school is
950 metres. Sarah leaves her house at 8:00 am and must be at school by 8:10 am.
Will Sarah arrive to school on time? Justify your answer.
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Question 39
[Maximum mark: 6]
Jack rides his bike to work each morning. During the first minute, he travels 160 metres. In each subsequent minute, he
travels 80 % of the distance travelled during the previous minute.
The distance from his home to work is 750 metres. Jack leaves his house at 8:30 am and must be at work at 8:40 am.
Will Jack arrive to work on time? Justify your answer.
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MEDIUM
[Maximum mark: 5]
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IB Math AA HL Questionbank - Sequences & Series
The third term of an arithmetic sequence is equal to 7 and the sum of the first 8 terms is 20.
Find the common difference and the first term.
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Question 41
[Maximum mark: 6]
The first term and the common ratio of a geometric series are denoted, respectively, by u1 and r, where u1 , r ∈ Q. Given
that the fourth term is 64 and the sum to infinity is 625, find the value of u1 and the value of r.
​
​
​
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Question 42
[Maximum mark: 6]
The seventh term of an arithmetic sequence is equal to 1 and the sum of the first 16 terms is 52.
Find the common difference and the first term.
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Question 43
[Maximum mark: 6]
The sum of an infinite geometric sequence is 27. The second term of the sequence is 6. Find the possible values of r.
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IB Math AA HL Questionbank - Sequences & Series
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Question 44
[Maximum mark: 13]
The following table shows the price (y dollars) of six used trucks, depending on the distance they have travelled (x km).
The relationship between the variables can be modelled by the regression equation y = ax + b.
(a)
(i) Find the correlation coefficient.
(ii) Write down the value of a and the value of b.
[3]
(b) Use the regression equation to estimate the price of a truck that has travelled 160 000 km.
[2]
James buys the truck that has travelled 160 000 km, however doesn't drive it. The price of the truck decreases by 7%
each year it sits still in James' garage.
(c) Calculate the price of James' truck after 5 years.
[4]
(d) If James sells his truck for 20 000 dollars k years after he purchased it, find the value of k to the nearest year.
[4]
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Question 45
[Maximum mark: 7]
An infinite geometric series has u1 = k and u2 =
​
​
1 2
k − 2k , where k > 0.
2
​
(a) Find an expression for the common ratio, r, in terms of k .
[2]
(b) Find the values of k for which the sum to infinity of the series exists.
[2]
(c) Find the value of k when the sum of the infinite sequence is S∞ = 46.
[3]
​
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Question 46
[Maximum mark: 8]
In an arithmetic sequence, u1 = logk (ab), u2 = logk (b), where k > 1 and a, b > 0.
​
​
​
​
(a) Show that d = − logk (a).
[2]
​
10
[6]
(b) Let a = k 4 and b = k 16 . Find the value of ∑ un .
​
​
n=1
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Question 47
[Maximum mark: 6]
The 1st, 5th and 13th terms of an arithmetic sequence, with common difference d, d =
 0, are the first three terms of a
geometric sequence, with common ratio r, r =
 1. Given that the 1st term of both sequences is 12, find the value of d and
the value of r.
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MEDIUM
Question 48
[Maximum mark: 6]
The sum of the first three terms of a geometric sequence is 81.3, and the sum of the infinite sequence is 300. Find the
common ratio.
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MEDIUM
Question 49
[Maximum mark: 8]
It is known that the number of trees in a small forest will decrease by 5 % each year unless some new trees are planted.
At the end of each year, 600 new trees are planted to the forest. At the start of 2021, there are 8200 trees in the forest.
(a) Show that there will be roughly 9060 trees in the forest at the start of 2026.
[4]
(b) Find the approximate number of trees in the forest at the start of 2041.
[4]
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Question 50
[Maximum mark: 13]
(a) The following diagram shows [PQ], with length 4 cm. The line is divided into an infinite number of line segments.
The diagram shows the first four segments.
The length of the line segments are m cm, m2 cm, m3 cm, …, where 0 < m < 1.
Show that m =
4
.
5
[5]
​
(b) The following diagram shows [RS], with length l cm, where l > 1. Squares with side lengths n cm, n2 cm, n3 cm, …
, where 0 < n < 1, are drawn along [RS]. This process is carried on indefinitely. The diagram shows the first four
squares.
The total sum of the areas of all the squares is
25
. Find the value of l.
11
​
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Formula Booklet
IB Math AA HL Questionbank - Sequences & Series
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Question 51
[Maximum mark: 15]
The first three terms of an infinite geometric sequence are k − 4, 4, k + 2, where k ∈ Z.
(a)
(i) Write down an expression for the common ratio, r.
(ii) Hence show that k satisfies the equation k 2 − 2k − 24 = 0.
(b)
[5]
(i) Find the possible values for k .
(ii) Find the possible values for r.
[5]
(c) The geometric sequence has an infinite sum.
(i) Which value of r leads to this sum. Justify your answer.
(ii) Find the sum of the sequence.
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[5]
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Question 52
[Maximum mark: 8]
Let f (x) = e3 sin( 4 ) , for x > 0.
πx
​
The k th maximum point on the graph of f has x-coordinate xk , where k ∈ Z+ .
​
(a) Given that xk+1 = xk + d, find d.
​
[4]
​
n
[4]
(b) Hence find the value of n such that ∑ xk = 992.
​
​
k=1
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IB Math AA HL Questionbank - Sequences & Series
Question 53
MEDIUM
[Maximum mark: 14]
Alex and Julie each have a goal of saving $30 000 to put towards a house deposit. They each have $16 000 to invest.
(a) Alex chooses his local bank and invests his $16 000 in a savings account that offers an interest rate of 5% per
annum compounded annually.
(i) Find the value of Alex's investment after 7 years, to the nearest hundred dollars.
(ii) Alex reaches his goal after n years, where n is an integer. Determine the value of n.
[4]
(b) Julie chooses a different bank and invests her $16 000 in a savings account that offers an interest rate of r% per
annum compounded monthly, where r is set to two decimal places.
Find the minimum value of r needed for Julie to reach her goal after 10 years.
[3]
(c) Xavier also wants to reach a savings goal of $30 000. He doesn't trust his local bank so he decides to put his money
into a safety deposit box where it does not earn any interest. His system is to add more money into the safety
deposit box each year. Each year he will add one third of the amount he added in the previous year.
(i) Show that Xavier will never reach his goal if his initial deposit into the safety deposit box is $16 000.
(ii) Find the amount Xavier needs to initially deposit in order to reach his goal after 7 years. Give your answer to
the nearest dollar.
[7]
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Question 54
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MEDIUM
[Maximum mark: 13]
Grant wants to save $40 000 over 5 years to help his son pay for his college tuition. He deposits $20 000 into a savings
account that has an interest rate of 6% per annum compounded monthly for 5 years.
(a) Show that Grant will not be able to reach his target.
[2]
(b) Find the minimum amount, to the nearest dollar, that Grant would need to deposit initially for him to reach his
target.
[3]
Grant only has $20 000 to invest, so he asks his sister, Caroline, to help him accelerate the saving process. Caroline is
happy to help and offers to contribute part of her income each year. Her annual income is $37 500 per year. She starts by
contributing one fifth of her annual income, and then decreases her contributions by half each year until the target is
reached. Caroline's contributions do not yield any interest.
(c) Show that Grant and Caroline together can reach the target in 5 years.
[4]
Grant and Caroline agree that Caroline should stop contributing once she contributes enough to complement the deficit
of Grant's investment.
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IB Math AA HL Questionbank - Sequences & Series
(d) Find the whole number of years after which Caroline will will stop contributing.
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Question 55
[Maximum mark: 6]
A bouncy ball is dropped from a height of 2 metres onto a concrete floor. After hitting the floor, the ball rebounds back
up to 80 % of it's previous height, and this pattern continues on repeatedly, until coming to rest.
(a) Show that the total distance travelled by the ball until coming to rest can be expressed by
2 + 4(0.8) + 4(0.8)2 + 4(0.8)3 + ⋯
[2]
(b) Find an expression for the total distance travelled by the ball, in terms of the number of bounces, n.
[2]
(c) Find the total distance travelled by the ball.
[2]
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Question 56
[Maximum mark: 7]
The sides of a square are 8 cm long. A new square is formed by joining the midpoints of the adjacent sides and two of the
resulting triangles are shaded as shown. This process is repeated 5 more times to form the right hand diagram below.
(a) Find the total area of the shaded region in the right hand diagram above.
[4]
(b) Find the total area of the shaded region if the process is repeated indefinitely.
[3]
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IB Math AA HL Questionbank - Sequences & Series
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MEDIUM
Question 57
[Maximum mark: 14]
The first two terms of an infinite geometric sequence, in order, are
3 log3 x, 2 log3 x, where x > 0.
​
​
(a) Find the common ratio, r.
[2]
(b) Show that the sum of the infinite sequence is 9 log3 x.
[3]
​
The first three terms of an arithmetic sequence, in order, are
log3 x, log3
​
x
x
, log3 , where x > 0.
3
9
​
​
​
​
(c) Find the common difference d, giving your answer as an integer.
[3]
Let S6 be the sum of the first 6 terms of the arithmetic sequence.
​
(d) Show that S6 = 6 log3 x − 15.
[3]
(e) Given that S6 is equal to one third of the sum of the infinite geometric
sequence, find x, giving your answer in the form ap where a, p ∈ Z.
[3]
​
​
​
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Question 58
[Maximum mark: 9]
The sum of the first n terms of a geometric sequence is given by
n
Sn = ∑
​
r=1
​
4 2
( )
5 3
​
r
​
(a) Find the first term of the sequence u1 .
[2]
(b) Find S∞ .
[3]
​
​
(c) Find the least value of n such that S∞ − Sn < 0.002.
​
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[4]
​
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IB Math AA HL Questionbank - Sequences & Series
Ask Newton
MEDIUM
Question 59
[Maximum mark: 6]
Given a sequence of integers, between 20 and 300, which are divisible by 9.
(a) Find their sum.
[2]
(b) Express this sum using sigma notation.
[2]
An arithmetic sequence has first term −500 and common difference of 8. The sum of the first n terms of this sequence is
negative.
(c) Find the greatest value of n.
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[2]
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HARD
Question 60
[Maximum mark: 8]
The first three terms of a geometric sequence are ln x9 , ln x3 , ln x, for x > 0.
(a) Find the common ratio.
[3]
∞
[5]
(b) Solve ∑ 33−k ln x = 27.
​
k=1
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[Maximum mark: 15]
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IB Math AA HL Questionbank - Sequences & Series
The first two terms of an infinite geometric sequence are u1 = 20 and u2 = 16 sin2 θ, where 0 < θ < 2π , and θ =
 π.
​
(a)
​
(i) Find an expression for r in terms of θ.
(ii) Find the possible values of r.
[5]
100
.
3 + 2 cos 2θ
(b) Show that the sum of the infinite sequence is
[4]
​
(c) Find the values of θ which give the greatest value of the sum.
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[6]
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HARD
Question 62
[Maximum mark: 15]
Bill takes out a bank loan of $100 000 to buy a premium electric car, at an annual interest rate of 5.49%. The interest is
calculated at the end of each year and added to the amount outstanding.
(a) Find the amount of money Bill would owe the bank after 10 years. Give your answer to the nearest dollar.
[3]
To pay off the loan, Bill makes quarterly deposits of $P at the end of every quarter in a savings account, paying a
nominal annual interest rate of 3.2%. He makes his first deposit at the end of the first quarter after taking out the loan.
(b) Show that the total value of Bill's savings after 10 years is P [
1.00840 − 1
].
1.008 − 1
[3]
​
(c) Given that Bill's aim is to own the electric car after 10 years, find the value for P to the nearest dollar.
[3]
Melinda visits a different bank and makes a single deposit of $Q, the annual interest rate being 3.5%.
(d)
(i) Melinda wishes to withdraw $8000 at the end of each year for a period of n years. Show that an expression for
the minimum value of Q is
8000
8000
8000
8000
+
+
+⋯+
.
1.035 1.0352
1.0353
1.035n
​
​
​
​
(ii) Hence, or otherwise, find the minimum value of Q that would permit Melinda to withdraw annual amounts
of $8000 indefinitely. Give your answer to the nearest dollar.
Formula Booklet
Question 63
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IB Math AA HL Questionbank - Sequences & Series
[Maximum mark: 7]
The sum of the first two terms of a geometric series is 7 and the sum of the first six terms is 91.
(a) Show that the common ratio r satisfies r2 = 3.
[4]
(b) Given r = 3,
(i) find the first term;
​
(ii) find the sum of the first eight terms.
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[3]
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HARD
Question 64
[Maximum mark: 14]
The cubic polynomial equation x3 + bx2 + cx + d = 0 has three roots x1 , x2 and x3 . By expanding the product (x −
x1 )(x − x2 )(x − x3 ), show that
​
​
(a)
​
​
​
​
(i) b = −(x1 + x2 + x3 );
​
​
​
(ii) c = x1 x2 + x1 x3 + x2 x3 ;
​
​
​
​
​
​
(iii) d = −x1 x2 x3 .
​
​
[3]
​
It is given that b = −9 and c = 45 for parts (b) and (c) below.
(b)
(i) In the case that the three roots x1 , x2 and x3 form an arithmetic
sequence, show that one of the roots is 3.
​
​
​
(ii) Hence determine the value of d.
[5]
(c) In another case the three roots form a geometric sequence. Determine
the value of d.
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