AIHL-SEQUENCES
Q1.[maximum marks:6] (Calculator)
Only one of the following four sequences is arithmetic and only one of them is geometric.
𝑎𝑛 = 1,5,10,15, … 𝑐𝑛 = 1.5,3,4.5,6, …
1 2 3 4 1 1
𝑏𝑛 = , , , , … 𝑑𝑛 = 2,1, , , …
2 3 4 5 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]
Q2. [maximum marks:6] (Calculator)
Only one of the following four sequences is arithmetic and only one of them is geometric.
1 1 1 1 1 1
𝑎𝑛 = , , , , … 𝑐𝑛 = 3,1, , , …
3 4 5 6 3 9
𝑏𝑛 = 2.5,5,7.5,10, … 𝑑𝑛 = 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]
Q3. [maximum marks:6] (Calculator)
An arithmetic sequence has 𝑢1 = 40, 𝑢2 = 32, 𝑢3 = 24
(a) Find the common difference, 𝑑. [2]
(b) Find 𝑢8 . [2]
(c) Find 𝑆8 . [2]
Q4. [maximum marks:6] (Calculator)
An arithmetic sequence has 𝑢1 = 12, 𝑢2 = 21, 𝑢3 = 30.
(a) Find the common difference, 𝑑. [2]
(b) Find 𝑢10 . [2]
(c) Find 𝑆10. [2]
Q5. [maximum marks:6] (Calculator)
The 15th term of an arithmetic sequence is 21 and the common difference is −4.
(a) Find the first term of the sequence.
[2]
(b) Find the 29th term of the sequence. [2]
(c) Find the sum of the first 40 terms of the sequence. [2]
Q6. [maximum marks:6] (Calculator)
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.
(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]
Q7. [maximum marks:6] (Calculator)
A geometric sequence has 20 terms, with the first four terms given below.
418.5,279,186,124, …
(a) Find 𝑟, the common ratio of the sequence. Give your answer as a fraction. [1]
(b) Find 𝑢5 , the fifth term of the sequence. Give your answer as a fraction.
[1]
(c) Find the smallest term in the sequence that is an integer.
[2]
(d) Find 𝑆10, the sum of the first 10 terms of the sequence. Give your answer correct to one
decimal place. [2]
Q8. [maximum marks:6] (Calculator)
Emily starts reading Leo Tolstoy's War and Peace on the 1st of February. The number of
pages she reads each day increases by the same number on each successive day.
Date in February 1 2 3 4 ⋯⋯⋯
Number of pages read 8 11 14 17 ………
(a) Calculate the number of pages Emily reads on the 14th of February. [3]
(b) Find the exact total number of pages Emily reads in the 28 days of February. [3]
Q9. [maximum marks:6] (Calculator)
The table shows the first four terms of three sequences: 𝑢𝑛 , 𝑣𝑛 , and 𝑤𝑛 .
𝑛
1 2 3 4
𝑢𝑛 12 24 48 72
𝑣𝑛 12 24 36 48
𝑤𝑛 12 24 48 96
(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]
Q10. [maximum marks:6] (Calculator)
Consider the infinite geometric sequence 3510, −2340,1560, −1040, …
(a) Find the common ratio. [2]
(b) Find the 12th term. [2]
(c) Find the exact sum of the infinite sequence. [2]
Q11. [maximum marks:6] (Calculator)
Consider the infinite geometric sequence 9000, −7200,5760, −4608, …
(a) Find the common ratio. [2]
(b) Find the 25 th term.
[2]
(c) Find the exact sum of the infinite sequence. [2]
Q12. [maximum marks:6] (Calculator)
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 60 th 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. [2]
Q13. [maximum marks:6] (Calculator)
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. [2]
Q14. [maximum marks:6] (Calculator)
The fifth term, 𝑢4 , of a geometric sequence is 135 . The sixth 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]
Q15. [maximum marks:6] (Calculator)
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]
Q16. [maximum marks:6] (Calculator)
The fifth term, 𝑢5 , of an arithmetic sequence is 5 . The eighth term, 𝑢8 , of the same sequence
is 14 .
(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]
Q17. [maximum marks:6] (Calculator)
On the first day of September, 2019 , Gloria planted 5 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 63 .
(a) Find the common difference of the sequence. [2]
(b) Find the total number of flowers Gloria is going to plant during September. [2]
(c) Gloria estimated she would plant 1000 flowers in the month of September. Calculate the
percentage error in Gloria's estimate. [2]
Q18. [maximum marks:6] (Calculator)
The fifth term, 𝑢5 , of a geometric sequence is 375 . The sixth term, 𝑢6 , of the sequence is 75
(a) Write down the common ratio of the sequence. [1]
(b) Find 𝑢1 . [2]
The sum of the first 𝑘 terms in the sequence is 292968 .
(c) Find the value of 𝑘. [3]
Q19. [maximum marks:6] (Calculator)
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:
(i) 10 years;
(ii) 20 years. [4]
(b) Find the number of years it will take for the population to reach 15000. [2]
Q20. [maximum marks:6] (Calculator)
A 3D printer builds a set of 49 Eiffel Tower Replicas in different sizes. The height of the
largest tower in this set is 64 cm. The heights of successive smaller towers are 95% of the
preceding larger tower, as shown in the diagram below.
(a) Find the height of the smallest tower in this set. [3]
(b) Find the total height if all 49 towers were placed one on top of another. [3]
Q21. [maximum marks:6] (Calculator)
The second and the third terms of a geometric sequence are 𝑢2 = 3 and 𝑢3 = 6.
(a) Find the value of 𝑟, the common ratio of the sequence. [2]
(b) Find the value of 𝑢6 . [2]
(c) Find the largest value of 𝑛 for which 𝑢𝑛 is less than 104 .
[2]
Q22. [maximum marks:6] (Calculator)
The Australian Koala Foundation estimates that there are about 45000 koalas left in the wild
in 2019. A year before, in 2018 , the population of koalas was estimated as 50000 .
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]
Q23. [maximum marks:12] (Calculator)
The sum of the first 𝑛 terms of an arithmetic sequence, 𝑆𝑛 = 𝑢1 + 𝑢2 + 𝑢3 + ⋯ + 𝑢𝑛 , is
given by 𝑆𝑛 = 2𝑛2 + 𝑛.
(a) Write down the values of 𝑆1 and 𝑆2 . [2]
(b) Write down the values of 𝑢1 and 𝑢2 . [2]
(c) Find 𝑑, the common difference of the sequence. [1]
(d) Find 𝑢10 , the tenth term of the sequence.
[2]
(e) Find the greatest value of 𝑛, for which 𝑢𝑛 is less than 100 . [3]
(f) Find the value of 𝑛, for which 𝑆𝑛 is equal to 1275 .
[2]
Q24. [maximum marks:6] (Calculator)
A battalion is arranged, per row, according to an arithmetic sequence. There are 24 troops in
the third row and 42 troops in the sixth row.
(a) Find the first term and the common difference of this arithmetic sequence. [3]
There are 15 rows in the battalion.
(b) Find the total number of troops in the battalion. [3]
Q25. [maximum marks:15] (Calculator)
Charles has a New Years Resolution that he wants to be able to complete 100 pushups in one
go without a break. He sets out a training regime whereby he completes 20 pushups on the
first day, then adds 5 pushups each day thereafter.
(a) Write down the number of pushups Charles completes
(i) on the 4th training day;
(ii) on the 𝑛th training day.
[3]
On the 𝑘 th training day Charles completes 100 pushups for the first time.
(b) Find the value of 𝑘. [2]
(c) Calculate the total number of pushups Charles completes on the first 10 training days.[4]
Charles is also working on improving his long distance swimming in preparation for an Iron
Man event in 12 week’s time. He swims a total of 10000 metres in the first week, and plans
to increase this by 10% each week up until the event.
(d) Find the distance Charles swims in the 6th week of training. [3]
(e) Calculate the total distance Charles swims until the event. [3]
Q26. [maximum marks:16] (Calculator)
The number of seats in an auditorium follows a regular pattern where the first row has 𝑢1
seats, and the amount increases by the same amount, 𝑑, each row. In the fifth row there are 62
seats and in the thirteenth row there are 86 seats.
(a) Write down an equation, in terms of 𝑢1 and 𝑑, for the amount of seats
(i) in the fifth row;
(ii) in the thirteenth row. [2]
(b) Find the value of 𝑢1 and 𝑑. [2]
(c) 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 3%
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.
(d)
(i) Find the price of the ticket for a seat in the fifth row, rounding your answer to two
decimal places.
(ii) Find the row of the seat at which the price of a ticket first falls below $70.
(iii) 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]
Q27. [maximum marks:14] (Calculator)
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 4 metres during the first second, 12 metres during the next second, 20
metres during the third second and continues in this way. The distances that the purse falls
during each second forms an arithmetic sequence.
(a)
(i) Write down the common difference, 𝑑, of this arithmetic sequence.
(ii) Write down the distance the purse falls during the fourth second. [2]
(b) Calculate the distance the purse falls during the 13th second. [2]
(c) Calculate the total distance the purse falls in the first 13 seconds. [2]
Georgia drops the purse from a height of 1250 metres above the ground.
(d) 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 2019 and in the first year
there were 20000 visitors. The number of people who visit the national park is expected to
increase by 8% each year.
(e) Calculate the number of people expected to visit the national park in 2020 . [2]
(f) Calculate the total number of people expected to visit the national park by the end of 2028
. [3]
Q28. [maximum marks:15] (Calculator)
A ball is dropped from the top of the Eiffel Tower, 324 metres from the ground. The ball falls
a distance of 4.9 metres during the first second, 14.7 metres during the next second, 24.5
metres during the third second, and so on. The distances that the ball falls each second form
an arithmetic sequence.
(a)
(i) Find 𝑑, the common difference of the sequence.
(ii) Find 𝑢5 , the fifth term of the sequence. [2]
(b) Find 𝑆6 , the sum of the first 6 terms of the sequence. [2]
(c) 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,
(d) find the distance the ball travels from the start of the 10th second to the end of the 15th
second. [3]
The Eiffel Tower in Paris, France was opened in 1889 , and 1.9 million visitors ascended it
during that first year. The number of people who visited the tower the following year (1890)
was 2 million.
(e) Calculate the percentage increase in the number of visitors from 1889 to 1890 . Give your
answer correct to one decimal place. [2]
(f) Use your answer to part (e) to estimate the number of visitors in 1900 , assuming that the
number of visitors continues to increase at the same percentage rate. [3]
Q29. [maximum marks:6] (Calculator)
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.
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]
Q30. [maximum marks:6] (Calculator)
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]
(c) Calculate the total height of swings if Peter is left to swing until coming to rest.[2]
Q31. [maximum marks:6] (Calculator)
Melinda has $300000 in a private foundation. Each year she donates 10% of the money
remaining in her private foundation to charity.
(a) Find the maximum number of years Melinda can donate to charity while keeping at least
$100000 in the private foundation. [3]
Bill invests $400000 in a bank account that pays a nominal interest rate of 4%, compounded
quarterly, for ten years.
(b) Calculate the value of Bill's investment at the end of this time. Give your answer correct
to the nearest dollar. [3]
Q32. [maximum marks:6] (Calculator)
Landmarks are placed along the road from London to Edinburgh and the distance between
each landmark is 16.1 km. The first milestone placed on the road is 124.7 km from London,
and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is
667.1 km.
(a) Find the distance between the fifth milestone and London. [3]
(b) Determine how many milestones there are along the road. [3]
Q33. [maximum marks:6] (Calculator)
Consider the sum 𝑆 = ∑𝑙𝑘=4 (2𝑘 − 3), where 𝑙 is a positive integer greater than 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]
Q34. [maximum marks:6] (Calculator)
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+ .
(b) Find the value of the sum of the geometric series ∑6𝑘=1 𝑣𝑘 . [2]
Q35. [maximum marks:6] (Calculator)
Let 𝑢𝑛 = 4𝑛 + 1, for 𝑛 ∈ ℤ+ .
(a)
(i) Using sigma notation, write down an expression for 𝑢1 + 𝑢2 + 𝑢3 + ⋯ + 𝑢20 .
(ii) Find the value of the sum from part (a) (i).
[4]
A geometric sequence is defined by 𝑣𝑛 = 9 × 4𝑛−1 , for 𝑛 ∈ ℤ+ .
(b) Find the value of the sum of the geometric series ∑5𝑘=1 𝑣𝑘 . [2]
Q36. [maximum marks:6] (Calculator)
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. [3]
(b) Find the total area of the shaded region if the process is repeated indefinitely. [3]
Q37. [maximum marks:7] (Calculator)
The half-life, 𝑇, in years, of a radioactive isotope can be modelled by the function
ln 0.5
𝑇(𝑘) = , 0 < 𝑘 < 100
𝑘
ln (1 − 100)
where 𝑘 is the decay rate, in percent, per year of the isotope.
(a) The decay rate of Hydrogen-3 is 5.5% per year. Find its half-life. [2]
The half-life of Uranium-232 (U-232) is 68.9 years. A sample containing 250 grams of U-
232 is obtained and stored as a side product of a nuclear fuel cycle.
(b) Find the decay rate per year of U-232. [2]
(c) Find the amount of U-232 left in the sample after:
(i) 68.9 years;
(ii) 100 years. [3]
Q38. [maximum marks:7] (Calculator)
The half-life, 𝑇, in days, of a radioactive isotope can be modelled by the function
ln 0.5
𝑇(𝑘) = , 0 < 𝑘 < 100
𝑘
ln (1 − 100)
where 𝑘 is the decay rate, in percent, per day of the isotope.
(a) The decay rate of Gold-196 is 6.2% per day. Find its half-life. [2]
The half-life of Phosphorus-32 (P-32) is 14.3 days. A sample containing 120 grams of P-32
is produced and stored in a biochemistry laboratory.
(b) Find the decay rate per day of P-32.
[2]
(c) Find the amount of P-32 left in the sample after:
(i) 14.3 days;
(ii) 50 days. [3]
Q39. [maximum marks:15] (Calculator)
Consider the sequence 𝑢1 , 𝑢2 , 𝑢3 , … , 𝑢𝑛 , … where
𝑢1 = 860, 𝑢2 = 980, 𝑢3 = 1100, 𝑢4 = 1220
The sequence continues in the same manner.
(a) Find the value of 𝑢50 .
[3]
(b) Find the sum of the first 10 terms of the sequence. [3]
Now consider the sequence 𝑣1 , 𝑣2 , 𝑣3 , … , 𝑣𝑛 , … where
𝑣1 = 2, 𝑣2 = 4, 𝑣3 = 8, 𝑣4 = 16
This sequence continues in the same manner.
(c) Find the exact value of 𝑣13 .
[3]
(d) Find the sum of the first 10 terms of this sequence. [3]
𝑘 is the smallest value of 𝑛 for which 𝑣𝑛 is greater than 𝑢𝑛 .
(e) Calculate the value of 𝑘. [3]
Q40. [maximum marks:19] (Calculator)
On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in 25
seconds. Each lap Eddy cycles takes him 1.6 seconds longer than the previous lap.
(a) Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]
Eddy cycles his last lap in 55.4 seconds.
(b) Find how many laps he has cycled on Wednesday.
[3]
(c) 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
25 seconds. Each lap Mario cycles takes him 1.05 times as long as his previous lap.
(d) Find the time, in seconds, Mario takes to cycle his fifth lap. [3]
(e) Find the total time, in minutes, Mario takes to cycle his first ten laps. [3]
Each lap Eddy cycles again takes him 1.6 seconds longer that his previous lap. After a certain
number of laps Eddy takes less time per lap than Mario.
(f) Find the number of the lap when this happens. [3]
Q41. [maximum marks:6] (Calculator)
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]
Q42. [maximum marks:6] (Calculator)
A bouncy ball is dropped out of a second story classroom window, 5 m off the ground. Every
time the ball hits the ground it bounces 89% of its previous height.
(a) Find the height the ball reaches after the 11th bounce. [3]
(b) Find the total distance travelled by the ball until it comes to rest. [3]
