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Arithmetic Sequences and Series · Paper 2 — Free Maths AI SL Practice Questions

1MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A new smartphone app records the number of push-ups a user completes each day. On the first day, the user completes 1212 push-ups. Each subsequent day, the user completes 55 more push-ups than the previous day. The number of push-ups completed on day nn is denoted by unu_n.
(a)
Write down an expression for unu_n in terms of nn. [1 mark]
(b)
Calculate the number of push-ups the user completes on day 2020. [2 marks]
(c)
Determine the first day on which the user completes at least 100100 push-ups. [3 marks]
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2MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A theme park has a queue for a new ride. On the first day, 120120 people queue. The park managers predict that each subsequent day, the number of people queuing will increase by a constant amount. On the 5th day, 200200 people queue.
(a)
Calculate the common difference in the number of people queuing each day. [2 marks]
(b)
Calculate the number of people predicted to queue on the 12th day. [2 marks]
(c)
The ride can accommodate a maximum of 500500 people per day. Determine the first day on which the predicted number of people queuing exceeds this capacity. [2 marks]
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3MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A biologist is studying the growth of a bamboo plant. On the first day of measurement, the plant is 25 cm25\,\text{cm} tall. The biologist models the height as an arithmetic sequence, where the plant grows by a constant amount each day.
(a)
On the 8th day, the plant is 74 cm74\,\text{cm} tall. Calculate the daily growth rate. [2 marks]
(b)
Calculate the height of the plant on the 15th day. [2 marks]
(c)
A second bamboo plant also starts at 25 cm25\,\text{cm} on day 1 and grows according to the model h(n)=25×(1.09)n−1h(n) = 25 \times (1.09)^{n-1}, where nn is the day number. Determine the first day on which the second plant is taller than the first plant, and evaluate whether the arithmetic model or the exponential model predicts a greater height on day 30. [2 marks]
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4MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A construction company is building a staircase. The first step has a height of 12 cm12\,\text{cm}. Each subsequent step is 2 cm2\,\text{cm} higher than the previous step.
(a)
Show that the heights of the steps form an arithmetic sequence with common difference d=2 cmd = 2\,\text{cm}. [1 mark]
(b)
Calculate the height of the 1515th step. [2 marks]
(c)
The total combined height of all steps must equal exactly 620 cm\text{620}\,\text{cm}. Determine the number of steps in the staircase. [3 marks]
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5MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A biologist is studying a population of bacteria in a petri dish. At 08:00, there are 500500 bacteria. The number of bacteria increases by a fixed amount each hour. At 11:00, there are 800800 bacteria.
(a)
Calculate the common difference of this arithmetic sequence. [2 marks]
(b)
Calculate the number of bacteria at 15:00. [2 marks]
(c)
The petri dish has a maximum capacity of 20002000 bacteria. The biologist claims the dish will reach capacity before 20:00 on the same day. Determine whether the biologist's claim is correct, showing all working. [2 marks]
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6MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A construction company is building a tower of concrete blocks. The base layer (layer 1) contains 2424 blocks. Each subsequent layer contains 33 fewer blocks than the layer below it. The top layer contains exactly 33 blocks.
(a)
Determine the number of layers in the tower. [3 marks]
(b)
Calculate the total number of blocks used to build the tower. [2 marks]
(c)
Each concrete block has a mass of 12.5 kg12.5\,\text{kg}. The construction company can transport a maximum of 800 kg800\,\text{kg} of blocks per trip. Determine the minimum number of trips required to transport all the blocks needed to build the tower. [1 mark]
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7MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A gardener plants trees in a straight line. The first tree is planted 55 metres from a fence. Each subsequent tree is planted 33 metres further from the fence than the previous tree. The distance from the fence to the nthn^{\text{th}} tree is unu_n metres. un=u1+(n−1)du_n = u_1 + (n-1)d
(a)
Write down the value of u1u_1 and the value of dd. [1 mark]
(b)
Calculate the distance from the fence to the 12th12^{\text{th}} tree. [2 marks]
(c)
The last tree is 6262 metres from the fence. The gardener claims the total distance of all trees from the fence exceeds 650650 metres. Determine whether the gardener's claim is correct. [3 marks]
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8MasterySAQ-SDefinition and general term of arithmetic sequences8 marksPaper 2~12 min
A construction company is building a staircase. The first step has a height of 12 cm12\,\text{cm}. Each subsequent step has a height that is 3 cm3\,\text{cm} more than the previous step. The staircase has 2020 steps.
(a)
Show that the height of the nnth step is given by un=3n+9 cmu_n = 3n + 9\,\text{cm}. [2 marks]
(b)
Calculate the height of the 1515th step. [2 marks]
(c)
The total height of the staircase is the sum of the heights of all 2020 steps. A building regulation states that the total height of a staircase must not exceed 4.5 m4.5\,\text{m}. Determine whether this staircase satisfies the regulation. [4 marks]
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9MasterySAQ-SDefinition and general term of arithmetic sequences9 marksPaper 2~14 min
A biologist is studying the growth of a bamboo plant. On the first day of measurement, the plant is 45 cm45\,\text{cm} tall. Each subsequent day, it grows by a constant amount. On the 10th day, the plant is 108 cm108\,\text{cm} tall.
(a)
Find the common difference, dd, for the daily height increase of the plant. [3 marks]
(b)
Hence, calculate the height of the plant on the 20th day. [2 marks]
(c)
A greenhouse has a maximum internal height of 2.50 m2.50\,\text{m}. Determine on which day the plant first exceeds this limit, and hence evaluate whether the plant will fit inside the greenhouse for at least three weeks from the first day of measurement. [4 marks]
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10MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min
A construction company is building a new office tower. The ground floor is at a height of 4 m4\,\text{m} above street level. Each subsequent floor is 3.2 m3.2\,\text{m} higher than the floor directly below it. The heights of the floors above street level form an arithmetic sequence.
(a)
Write down the values of u1u_1 and dd for this sequence. [1 mark]
(b)
Calculate the height above street level of the 15th floor. [2 marks]
(c)
The construction crane can only reach floors whose height above street level is less than 100 m100\,\text{m}. Determine the maximum number of complete floors that can be built so that the crane can reach all of them. [3 marks]
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11MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A cinema has 1818 seats in the first row. Each subsequent row has 44 more seats than the row in front of it. The last row has 8282 seats.
(a)
Determine the number of rows in the cinema. [3 marks]
(b)
Calculate the total number of seats in the cinema. [3 marks]

Solutions

12MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min
A company is manufacturing cylindrical metal rods. The lengths of the rods (in cm) form an arithmetic sequence. The shortest rod has a length of 8 cm8\,\text{cm} and the longest rod has a length of 35 cm35\,\text{cm}. There are 1010 rods in total.
(a)
Calculate the common difference between consecutive rod lengths. [2 marks]
(b)
Calculate the total length of all 1010 rods placed end to end. The company receives an order for rods with a total combined length of exactly 306 cm306\,\text{cm}. The rods must form an arithmetic sequence with the same shortest rod length of 8 cm8\,\text{cm} and the same common difference found in part (a). [2 marks]
(c)
Determine the number of rods required to achieve a total length of exactly 306 cm306\,\text{cm}. [2 marks]
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13MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

Data

Sn=n2(2u1+(n−1)d)S_n = \frac{n}{2}(2u_1 + (n-1)d)
A gardener plants trees in rows. The first row contains 1515 trees. Each subsequent row contains 22 more trees than the previous row. The gardener wishes to plant at least 510510 trees.
(a)
Show that the total number of trees planted in nn complete rows is given by Sn=n2+14nS_n = n^2 + 14n. [3 marks]
(b)
Hence, find the minimum number of complete rows needed to plant at least 510510 trees. [2 marks]
(c)
Determine the number of trees planted in the final (minimum) row, and hence evaluate whether the gardener plants exactly 510510 trees or more. [1 mark]
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14MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

Data

un=u1+(n−1)du_n = u_1 + (n-1)d
A biologist is studying the growth of a bacterial colony. On day 1, the colony covers an area of 12 cm212\,\text{cm}^2. Each subsequent day, the area increases by a constant amount. On day 10, the area is 48 cm248\,\text{cm}^2.
(a)
Calculate the daily increase in area. [2 marks]
(b)
Determine on which day the area first exceeds 100 cm2100\,\text{cm}^2. [2 marks]
(c)
The biologist's petri dish has a total area of 200 cm2200\,\text{cm}^2. She claims the colony will fill the dish within 50 days of day 1. Determine whether claim is correct. [2 marks]

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15MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A cinema has 25 rows of seats. The first row has 18 seats. Each subsequent row has 4 more seats than the row in front of it. The number of seats in row nn is given by sns_n.
(a)
Write down an expression for sns_n in terms of nn. [1 mark]
(b)
Calculate the number of seats in row 15. [2 marks]
(c)
The cinema sells tickets for USD 12 per seat. Every seat in every row from row 1 to row kk is sold. Given that the total number of seats in rows 1 to kk must first exceed 800, find the value of kk and hence calculate the total ticket revenue. [3 marks]

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16MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A small business sells handmade candles. In the first month, they sell 8080 candles. The business plans to increase sales by a constant number of candles each month.
(a)
In the 44th month, they sell 116116 candles. Calculate the monthly increase in sales. [2 marks]
(b)
Calculate the number of candles predicted to be sold in the 1010th month. [2 marks]
(c)
Each candle sells for USD 4.50 and costs USD 2.10 to make. The business has fixed monthly costs of USD 312. Determine whether the business makes a profit in month 1616, showing all working. [2 marks]

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17MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min
A library has a membership drive. In the first month, 6060 new members join. The library expects the number of new members each month to increase by a constant amount dd.
(a)
In the 77th month, 108108 new members join. Calculate dd. [2 marks]
(b)
Calculate the number of new members predicted to join the 1818th month. [2 marks]
(c)
The library's goal is to reach a cumulative total of at least 30003000 new members. Determine the first month by the end of which this goal is met. [2 marks]

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18MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min
A small business owner saves money for equipment upgrades. In the first month she saves USD 500. Each subsequent month she increases her savings by a fixed amount dd compared to the previous month. After 24 months she has saved a total of USD 21 600.
(a)
Show that the amount saved in the 24th month is USD 1300. [3 marks]
(b)
Determine the first month in which her monthly savings exceed USD 1000. [3 marks]
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19MasterySAQ-SSum of an arithmetic sequence9 marksPaper 2~14 min

Data

un=u1+(n−1)du_n = u_1 + (n-1)d, Sn=n2(u1+un)\quad S_n = \dfrac{n}{2}(u_1 + u_n)
A theatre has seats arranged in rows. The first row has 2424 seats. Each subsequent row has 22 more seats than the previous row. The last row has 7474 seats.
(a)
Determine the number of rows in theatre. [3 marks]
(b)
Calculate the total number of seats in theatre. [2 marks]
(c)
The theatre manager claims that more than 60%60\% of the total seats are in the last 1010 rows. Determine whether this claim is correct, showing all working. [4 marks]

Solutions

20MasterySAQ-SSum of an arithmetic sequence8 marksPaper 2~12 min
A ball is dropped from a height of 200 cm200\,\text{cm}. On each bounce, it rises to a height that is 15 cm15\,\text{cm} less than the previous height. The ball is caught after it rises on the 10th bounce.
(a)
Calculate the height the ball reaches on the 10th bounce. [2 marks]
(b)
Show that the total vertical distance travelled by the ball, from the moment is dropped until it is caught, is 2500 cm2500\,\text{cm}. [4 marks]
(c)
The ball is instead dropped from a height of h cmh\,\text{cm}, with all other conditions unchanged. Given that the ball must still complete 10 bounces before being caught (i.e. the height on the 10th bounce must be positive), determine the range of values of hh. [2 marks]
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