IB Mathematics AI SL - Questionbank
Sequences & Series
Arithmetic & Geometric Sequences & Series, Finding Terms & Sum of Terms, Sigma Notation, Applications...
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Question 1
[Maximum mark: 6]
The th term of an arithmetic sequence is and the common difference is .
-
Find the first term of the sequence. [2]
-
Find the th term of the sequence. [2]
-
Find the sum of the first terms of the sequence. [2]
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Question 2
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]
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Question 3
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [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.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
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Question 5
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
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Question 6
[Maximum mark: 7]
Brendan is training for a long distance bike race.
In week of his training he cycled km. In week he cycled km. This pattern continues, with him cycling an extra km per week.
The distances Brendan cycled in the first weeks of training is shown in the following table.
-
Calculate how far he cycles in the th week of his training. [3]
-
In total how far has he cycled after weeks? [2]
-
Find the mean distance per week he cycled over 17 weeks. [2]
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Question 7
[Maximum mark: 6]
A geometric sequence has , and .
-
Find the common ratio, . [2]
-
Find the exact value of . [2]
-
Find the exact value of . [2]
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Question 8
[Maximum mark: 6]
Emily starts reading Leo Tolstoy's War and Peace on the st of February. The number of pages she reads each day increases by the same number on each successive day.
-
Calculate the number of pages Emily reads on the th of February. [3]
-
Find the exact total number of pages Emily reads in the days of February.[3]
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Question 9
[Maximum mark: 6]
A geometric sequence has terms, with the first four terms given below.
-
Find , the common ratio of the sequence. Give your answer as a fraction. [1]
-
Find , the fifth term of the sequence. Give your answer as a fraction. [1]
-
Find the smallest term in the sequence that is an integer. [2]
-
Find , the sum of the first terms of the sequence. Give your answer correct to one decimal place. [2]
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Question 10
[Maximum mark: 6]
A tennis ball bounces on the ground times. The heights of the bounces, form a geometric sequence. The height that the ball bounces the first time, , is cm, and the second time, , is cm.
-
Find the value of the common ratio for the sequence. [2]
-
Find the height that the ball bounces the tenth time, . [2]
-
Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to decimal places. [2]
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Question 11
[Maximum mark: 6]
The table shows the first four terms of three sequences: , , and .
-
State which sequence is
-
arithmetic;
-
geometric. [2]
-
-
Find the sum of the first terms of the arithmetic sequence. [2]
-
Find the exact value of the th term of the geometric sequence. [2]
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Question 12
[Maximum mark: 6]
The table below shows the first four terms of three sequences: , , and .
-
State which sequence is
-
arithmetic;
-
geometric. [2]
-
-
Find the exact value of the sum of the first terms of the arithmetic
sequence. [2]
-
Find the exact value of the th term of the geometric sequence. [2]
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Question 13
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
A population of goats on an island starts at . The population is
expected
to increase by % each year.
-
Find the expected population size after:
-
years;
-
years. [4]
-
-
Find the number of years it will take for the population to reach . [2]
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Question 14
[Maximum mark: 6]
On the first day of September, , Gloria planted flowers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowers she is going to plant in the last day of September is .
-
Find the common difference of the sequence. [2]
-
Find the total number of flowers Gloria is going to plant during September.[2]
-
Gloria estimated she would plant flowers in the month of September. Calculate the percentage error in Gloria's estimate. [2]
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Question 15
[Maximum mark: 6]
The fifth term, , of a geometric sequence is . The sixth term, , of the sequence is .
-
Write down the common ratio of the sequence. [1]
-
Find . [2]
The sum of the first terms in the sequence is .
- Find the value of . [3]
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Question 16
[Maximum mark: 6]
The third term, , of an arithmetic sequence is . The common
difference of
the sequence, , is .
-
Find , the first term of the sequence. [2]
-
Find , the th term of sequence. [2]
The first and fourth terms of this arithmetic sequence are the first two
terms
of a geometric sequence.
- Calculate the sixth term of the geometric sequence.
[2]
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Question 17
[Maximum mark: 6]
A D printer builds a set of Eifel Tower Replicas in different sizes. The height of the largest tower in this set is cm. The heights of successive smaller towers are % of the preceding larger tower, as shown in the diagram below.
-
Find the height of the smallest tower in this set. [3]
-
Find the total height if all towers were placed one on top of another. [3]
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Question 18
[Maximum mark: 6]
The fourth term, , of a geometric sequence is . The fifth term, , is .
-
Find the common ratio of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Calculate the sum of the first terms of the sequence. [2]
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Question 19
[Maximum mark: 6]
The fifth term, , of a geometric sequence is . The sixth term, , is .
-
Find the common ratio of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Calculate the sum of the first terms of the sequence. [2]
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Question 20
[Maximum mark: 6]
The fifth term, , of an arithmetic sequence is . The eighth term, , of the same sequence is .
-
Find , the common difference of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Find , the sum of the first terms of the sequence. [2]
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Question 21
[Maximum mark: 6]
The fifth term, , of an arithmetic sequence is . The eleventh term, , of the same sequence is .
-
Find , the common difference of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Find , the sum of the first terms of the sequence. [2]
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Question 22
[Maximum mark: 15]
Charles has a New Years Resolution that he wants to be able to complete pushups in one go without a break. He sets out a training regime whereby he completes pushups on the first day, then adds pushups each day thereafter.
- Write down the number of pushups Charles completes
-
on the th training day;
-
on the th training day. [3]
-
On the th training day Charles completes pushups for the first time.
-
Find the value of . [2]
-
Calculate the total number of pushups Charles completes on the first training days. [4]
Charles is also working on improving his long distance swimming in preparation for an Iron Man event in weeks time. He swims a total of metres in the first week, and plans to increase this by % each week up until the event.
-
Find the distance Charles swims in the th week of training. [3]
-
Calculate the total distance Charles swims until the event. [3]
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Question 23
[Maximum mark: 6]
The second and the third terms of a geometric sequence are and .
-
Find the value of , the common ratio of the sequence. [2]
-
Find the value of . [2]
-
Find the largest value of for which is less than .[2]
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Question 24
[Maximum mark: 12]
The sum of the first terms of an arithmetic sequence, , is given by .
-
Write down the values of and . [2]
-
Write down the values of and . [2]
-
Find , the common difference of the sequence. [1]
-
Find , the tenth term of the sequence. [2]
-
Find the greatest value of , for which is less than . [3]
-
Find the value of , for which is equal to . [2]
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Question 25
[Maximum mark: 6]
The Australian Koala Foundation estimates that there are about koalas left in the wild in . A year before, in , the population of koalas was estimated as . Assuming the population of koalas continues to decrease by the same percentage each year, find:
-
the exact population of koalas in ; [3]
-
the number of years it will take for the koala population to reduce to half of its number in . [3]
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Question 26
[Maximum mark: 6]
A battalion is arranged, per row, according to an arithmetic sequence. There are troops in the third row and troops in the sixth row.
-
Find the first term and the common difference of this arithmetic sequence. [3]
There are rows in the battalion.
- Find the total number of troops in the battalion.
[3]
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Question 27
[Maximum mark: 6]
Melinda has in a private foundation. Each year she donates of the money remaining in her private foundation to charity.
-
Find the maximum number of years Melinda can donate to charity while keeping at least in the private foundation. [3]
Bill invests in a bank account that pays a nominal interest rate of %, compounded quarterly, for ten years.
- Calculate the value of Bill's investment at the end of this
time. Give your answer correct to the nearest dollar.
[3]
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Question 28
[Maximum mark: 15]
Towards the end of 2004, a theatre company upgraded their auditorium and installed new comfortable ergonomic chairs for the audience.
After the redesign, there were seats in the first row and each subsequent row had three more seats than the previous row.
- If the auditorium had a total of rows, find
-
the total number of seats in the last row.
-
the total number of seats in the auditorium. [5]
-
The auditorium reopened for performances at the start of 2005. The average number of visitors per show during that year was . In 2006, the average number of visitors per show increased by .
- Find the average number of visitors per show in 2006. [1]
The average number of visitors per show continued to increase by each year.
- Determine the first year in which the total number of visitors to a
show exceeded the seating capacity of the auditorium. [5]
The theatre company hosts shows per year.
- Determine the total number of visitors that attended the auditorium
from when it opened in 2005 until the end of 2011. Round your answer
correct to the nearest integer. [4]
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Question 29
[Maximum mark: 6]
The first term of an arithmetic sequence is and the common difference is .
-
Find the value of the nd term of the sequence. [2]
The first term of a geometric sequence is . The th term of the geometric sequence is equal to the th term of the arithmetic sequence given above.
-
Write down an equation using this information. [2]
-
Calculate the common ratio of the geometric sequence. [2]
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Question 30
[Maximum mark: 15]
A ball is dropped from the top of the Eiffel Tower, metres from the ground. The ball falls a distance of metres during the first second, metres during the next second, metres during the third second, and so on. The distances that the ball falls each second form an arithmetic sequence.
-
-
Find , the common difference of the sequence.
-
Find , the fifth term of the sequence. [2]
-
-
Find , the sum of the first terms of the sequence. [2]
-
Find the time the ball will take to reach the ground. Give your answer in seconds correct to one decimal place. [3]
Assuming the ball is dropped another time from a much higher height than of the Eiffel Tower,
-
find the distance the ball travels from the start of the th second to the end of the th second. [3]
The Eiffel Tower in Paris, France was opened in , and million visitors ascended it during that first year. The number of people who visited the tower the following year () was million.
-
Calculate the percentage increase in the number of visitors from to . Give your answer correct to one decimal place. [2]
-
Use your answer to part (e) to estimate the number of visitors in , assuming that the number of visitors continues to increase at the same percentage rate. [3]
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Question 31
[Maximum mark: 13]
On September 1st, an orchard commences the process of harvesting hectares of apple trees. At the end of September 4th, there were hectares remaining to be harvested, and at the end of September 8th, there were hectares remaining. Assuming that the number of hectares harvested each day is constant, the total number of hectares remaining to be harvested can be described by an arithmetic sequence.
-
Find the number of hectares of apple trees that are harvested each day. [3]
-
Determine the number of hectares remaining to be harvested at the end of September 1st. [1]
-
Determine the date on which the harvest will be complete. [2]
In 2021 the orchard sold their apple crop for . It is expected that the selling price will then increase by annually for the next years.
-
Determine the amount of money the orchard will earn for their crop in 2026. Round your answer to the nearest dollar. [3]
-
-
Find the value of . Round your answer to the nearest integer.
-
Describe, in context, what the value in part (e) (i) represents. [3]
-
-
Comment on whether it is appropriate to model this situation in terms of a geometric sequence. [1]
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Question 32
[Maximum mark: 16]
The number of seats in an auditorium follows a regular pattern where the first row has seats, and the amount increases by the same amount, , each row. In the fifth row, there are seats and in the thirteenth row there are seats.
-
Write down an equation, in terms of and , for the amount of seats
-
in the fifth row.
-
in the thirteenth row.[2]
-
-
Find the value of and .[2]
-
Calculate the total number of seats if the auditorium has 20 rows.[3]
The cost of the ticket for a musical held at the auditorium is inversely proportional to the seat's row. The price for a seat in the first row is $120 dollars, and the price decreases each row. Thus, the value of the ticket for seats in the second row is $116.40 and $112.91 in the third one, etc.
-
-
Find the price of the ticket for a seat in the fifth row, rounding your answer to two decimal places.
-
Find the row of the seat at which the price of a ticket first falls below $70.
-
Find the total revenue the auditorium generates by tickets sales if 40 seats in each of the 20 rows are sold. Give your answer rounded to the nearest dollar.[9]
-
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Question 33
[Maximum mark: 7]
Two college students, David and Lisa, decide to invest money they have saved from working part-time jobs. David's investment strategy results in an increase of his investment amount by each year. Lisa's investment strategy results in her investment amount increasing by each year.
At the start of the second year of investing, David's total investment amount is and Lisa's is .
- Calculate
-
the original amount David invested.
-
the original amount Lisa invested.[4]
-
During a certain year, , Lisa's investment amount becomes larger than David's amount for the first time.
- Find the value of . [3]
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Question 34
[Maximum mark: 14]
Georgia is on vacation in Costa Rica. She is in a hot air balloon over a lush jungle in Muelle.
When she leans forward to see the treetops, she accidentally drops her purse. The purse falls down a distance of metres during the first second, metres during the next second, metres during the third second and continues in this way. The distances that the purse falls during each second forms an arithmetic sequence.
-
-
Write down the common difference, , of this arithmetic sequence.
-
Write down the distance the purse falls during the fourth second. [2]
-
-
Calculate the distance the purse falls during the th second. [2]
-
Calculate the total distance the purse falls in the first seconds. [2]
Georgia drops the purse from a height of metres above the ground.
-
Calculate the time, to the nearest second, the purse will take to reach
the ground. [3]
Georgia visits a national park in Muelle. It is opened at the start of and in the first year there were visitors. The number of people who visit the national park is expected to increase by each year.
-
Calculate the number of people expected to visit the national park in . [2]
-
Calculate the total number of people expected to visit the national park by the end of . [3]
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Question 35
[Maximum mark: 5]
A bouncy ball is dropped out of a second story classroom window, m off the ground. Every time the ball hits the ground it bounces % of its previous height.
-
Find the height the ball reaches after the nd bounce. [2]
-
Find the total distance the ball has travelled when it hits the ground for the th time. [3]
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Question 36
[Maximum mark: 6]
Let , for .
-
-
Using sigma notation, write down an expression for .
-
Find the value of the sum from part (a) (i). [4]
-
A geometric sequence is defined by , for .
- Find the value of the sum of the geometric series
.[2]
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Question 37
[Maximum mark: 6]
Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is km. The first milestone placed on the road is km from London, and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is km.
-
Find the distance between the fifth milestone and London. [3]
-
Determine how many milestones there are along the road. [3]
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Question 38
[Maximum mark: 6]
Let , for .
-
-
Using sigma notation, write down an expression for .
-
Find the value of the sum from part (a) (i). [4]
-
A geometric sequence is defined by , for .
- Find the value of the sum of the geometric series
.[2]
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Question 39
[Maximum mark: 6]
Consider the sum , where is a positive integer greater than .
-
Write down the first three terms of the series. [2]
-
Write down the number of terms in the series. [1]
-
Given that , find the value of . [3]
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Question 40
[Maximum mark: 15]
Consider the sequence where
The sequence continues in the same manner.
-
Find the value of . [3]
-
Find the sum of the first terms of the sequence. [3]
Now consider the sequence where
This sequence continues in the same manner.
-
Find the exact value of . [3]
-
Find the sum of the first terms of this sequence. [3]
is the smallest value of for which is greater than .
- Calculate the value of . [3]
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Question 41
[Maximum mark: 19]
On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in seconds. Each lap Eddy cycles takes him seconds longer than the previous lap.
-
Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]
Eddy cycles his last lap in seconds.
-
Find how many laps he has cycled on Wednesday. [3]
-
Find the total time, in minutes, cycled by Eddy on Wednesday. [4]
On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in seconds. Each lap Mario cycles takes him times as long as his previous lap.
-
Find the time, in seconds, Mario takes to cycle his fifth lap. [3]
-
Find the total time, in minutes, Mario takes to cycle his first ten laps. [3]
Each lap Eddy cycles again takes him seconds longer that his
previous lap.
After a certain number of laps Eddy takes less time per lap than Mario.
- Find the number of the lap when this happens. [3]
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Question 42
[Maximum mark: 7]
The half-life, , in years, of a radioactive isotope can be modelled by the function
where is the decay rate, in percent, per year of the isotope.
-
The decay rate of Hydrogen- is % per year. Find its half-life.[2]
The half-life of Uranium- (U-) is years. A sample containing grams of U- is obtained and stored as a side product of a nuclear fuel cycle.
-
Find the decay rate per year of U-. [2]
-
Find the amount of U- left in the sample after:
-
years;
-
years. [3]
-
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Question 43
[Maximum mark: 7]
The half-life, , in days, of a radioactive isotope can be modelled by the function
where is the decay rate, in percent, per day of the isotope.
-
The decay rate of Gold- is % per day. Find its half-life.[2]
The half-life of Phosphorus- (P-) is days. A sample containing grams of P- is produced and stored in a biochemistry laboratory.
-
Find the decay rate per day of P-. [2]
-
Find the amount of P- left in the sample after:
-
days;
-
days. [3]
-
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Question 44
[Maximum mark: 7]
Jenni is conducting an experiment with a spring and has attached a mass so that it will oscillate up and down.
She is measuring the -coordinate of the centre of the mass.
At the start of the experiment the mass is at rest with its centre being at the point .
She gives the mass a nudge upwards in the positive -direction. She makes her first measurement of when the centre of the mass is at the first maximum point (). The units of the -coordinate are in millimetres.
The mass then moves downwards passing the -axis and reaching its first minimum point (). Jenni makes her second measurement of the -coordinate of the centre of a the mass as .
The mass then moves up past the -axis to the next maximum point () and Jenni makes her third measurement of .
The diagram below shows how the mass moves up and down until Jenni makes her rd measurement.
Jenni notices that the -coordinates of the three measurements form a geometric sequence.
- Find . [2]
The spring continues to oscillate up and down with Jenni measuring the -coordinate in the same way as described.
The results continue to form a geometric sequence.
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Find the th term in the sequence. Give your answer to 3 decimal places. [2]
-
Show that the total distance travelled in the -direction by the mass when the th measurement is made is mm. [3]
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The IB Math Applications and Interpretation (AI) SL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AI SL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AI Standard Level course.
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