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Arithmetic and Geometric Sequences · Paper 2 — Free Maths AA SL Practice Questions

1MasterySAQ-SDefinition and general term of arithmetic sequencess8 marksPaper 2~12 min
A company manufactures cylindrical containers. The heights of the containers, measured in centimetres, form an arithmetic sequence. The third container has a height of 16 cm16\,\text{cm} and the seventh container has a height of 28 cm28\,\text{cm}.
(a)
Find the common difference and the height of the first container. [3 marks]
(b)
Find the height of the twelfth container. [2 marks]
(c)
The company needs the total height of the first nn containers to reach at least 500 cm500\,\text{cm}. Find the minimum value of nn. [3 marks]
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Solutions

2MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A farmer is planting trees in rows. The number of trees in each row forms an arithmetic sequence. The first row has 1515 trees. The fifth row has 2727 trees.
(a)
Show that the common difference of the sequence is 33. [2 marks]
(b)
Calculate the number of trees in the twelfth row. [2 marks]
(c)
The total number of trees planted in the first nn rows is 540540. Calculate the value of nn. [2 marks]
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3MasterySAQ-SDefinition and general term of arithmetic sequencess6 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 has a height 2 cm2\,\text{cm} greater than the previous step.
(a)
Calculate the height of the 25th step. [2 marks]
(b)
Show that the total vertical height of a staircase with 25 steps is exactly 900 cm900\,\text{cm}. [2 marks]
(c)
The company needs the total vertical height to exceed 900 cm900\,\text{cm}. Determine the minimum number of steps required. [2 marks]
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4MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A biologist monitors the growth of a bamboo plant. On the first day of observation, the plant has a height of 40 cm40\,\text{cm}. The plant grows by a constant amount each day. On the 1010th day of observation, the plant has a height of 103 cm103\,\text{cm}.
(a)
Calculate the daily growth rate of the bamboo plant. [2 marks]
(b)
Determine on which day the height of the plant first exceeds 200 cm200\,\text{cm}. [2 marks]
(c)
The biologist claims that the plant will reach a height of exactly 250 cm250\,\text{cm} on a particular day of observation. Determine whether this claim is correct, justifying your answer. [2 marks]
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5MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A gardener plants trees in a straight line along a path. The first tree is planted 5 m5\,\text{m} from the start of the path. Each subsequent tree is planted 3 m3\,\text{m} further from the start than the previous tree.
(a)
Calculate the distance from the start of the path to the 2020th tree. [2 marks]
(b)
The last tree is planted 152 m152\,\text{m} from the start of the path. Find the total number of trees planted. [2 marks]
(c)
A water pipe is to run from the start of the path to each tree and back again. Calculate the total length of pipe required to service all trees exactly once each. [2 marks]
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6MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A factory produces solar panels. On the first day of production, 1212 panels are made. The number of panels produced each day forms an arithmetic sequence with common difference dd. On the 1010th day, 3939 panels are produced.
(a)
Calculate the value of dd. [2 marks]
(b)
Find the number of panels produced on the 1515th day. [2 marks]
(c)
The factory's target is to produce a total of at least 10001000 panels within the first nn days. Find the minimum value of nn. [2 marks]
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7MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A gardener plants trees at intervals along a straight path. The distance between the first and second tree is 5 m5\,\text{m}. The distances between consecutive trees form an arithmetic sequence with common difference dd. The total distance from the first tree to the fifth tree is 38 m38\,\text{m}.
(a)
Write down the number of gaps between the first tree and the fifth tree. [1 mark]
(b)
Find the common difference dd. [2 marks]
(c)
Find the total distance from the first tree to the tenth tree. [3 marks]
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8MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A construction company is stacking bricks. The bottom row contains 5050 bricks. Each row above contains 44 fewer bricks than the row directly below it, forming an arithmetic sequence.
(a)
Calculate the number of bricks in the 88th row from the bottom. [2 marks]
(b)
The top row contains 66 bricks. Determine the total number of rows in the stack. [2 marks]
(c)
The company extends the stack by adding a new bottom row of 5454 bricks, keeping the same common difference. The top row still contains 66 bricks. Determine the total number of bricks in the new stack. [2 marks]
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9MasterySAQ-SDefinition and general term of arithmetic sequencess8 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 4 cm4\,\text{cm} greater than the previous step.
(a)
Calculate the height of the 1515th step. [2 marks]
(b)
The top step has a height of 240 cm240\,\text{cm}. Determine the number of steps in the staircase. [2 marks]
(c)
The total vertical rise of the staircase equals the sum of the heights of all steps. Calculate the total vertical rise, and hence determine whether a staircase with the same number of steps but a constant step height equal to the mean of the original step heights would have the same total vertical rise. [4 marks]
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10MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A staircase is built with concrete steps. The first step has a height of 18 cm18\,\text{cm}. Each subsequent step has a height that is 2 cm2\,\text{cm} less than the previous step, forming an arithmetic sequence.
(a)
Calculate the height of the 5th step. [2 marks]
(b)
The last step has a height of 4 cm4\,\text{cm}. Determine the total number of steps in the staircase. [2 marks]
(c)
A revised design requires exactly nn steps, where the first step has height 18 cm18\,\text{cm} and the heights decrease by a constant amount so that the final step has height 2 cm2\,\text{cm}. The total combined height of all steps must not exceed 120 cm120\,\text{cm}. Determine the maximum value of nn, and state the common difference for this value of nn. [2 marks]
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11MasterySAQ-SDefinition and general term of arithmetic sequencess8 marksPaper 2~12 min
A construction company stacks bricks in a pyramid. The top row has 11 brick. Each row below has 55 more bricks than the row above it. The bottom row has 9696 bricks.
(a)
Write down the value of the first term u1u_1 and the common difference dd of this arithmetic sequence. [1 mark]
(b)
Show that the number of bricks in the nnth row is un=5n−4u_n = 5n - 4. [2 marks]
(c)
Calculate the number of rows in the pyramid. [2 marks]
(d)
Calculate the total number of bricks in the pyramid. [2 marks] (e) The company has 950950 bricks available. Determine the number of complete rows they can build, justifying your answer. [3 marks]

Solutions

12MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min

Data

un=u1+(n−1)du_n = u_1 + (n-1)d, Sn=n2(2u1+(n−1)d)\quad S_n = \dfrac{n}{2}(2u_1 + (n-1)d)
The 5th term of an arithmetic sequence is 1717 and the 12th term is 3838.
(a)
(i) Find the common difference dd. [2]
(ii) Find the first term u1u_1. [1 mark]
(b)
Find the value of nn such that un=101u_n = 101. [2 marks]
(c)
The sum of the first kk terms of the sequence exceeds 20002000. Find the least possible integer value of kk. [3 marks]
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13MasterySAQ-SSum of an arithmetic sequences6 marksPaper 2~9 min
A cyclist trains on a straight road. On the first day, she cycles 8 km8\,\text{km}. Each subsequent day, she increases her distance by 2 km2\,\text{km} compared to the previous day. She trains for 2525 days.
(a)
Show that the distance cycled on the 2525th day is 56 km56\,\text{km}. [2 marks]
(b)
Calculate the total distance cycled over the 2525 days. [2 marks]
(c)
The cyclist wants to modify her training so that she cycles a total of at least 1000 km1000\,\text{km} over 2525 days, keeping the common difference at 2 km2\,\text{km} per day. Determine the minimum distance, in km, she must cycle on the first day. [2 marks]
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Solutions

14MasterySAQ-SDefinition and general term of arithmetic sequencess7 marksPaper 2~11 min

Data

un=u1+(n−1)dSn=n2(2u1+(n−1)d)u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(2u_1 + (n-1)d)
The 3rd term of an arithmetic sequence is 1010 and the 8th term is 2525.
(a)
Calculate the first term u1u_1 and the common difference dd. [3 marks]
(b)
Calculate the sum of the first 1515 terms. [2 marks]
(c)
Determine the smallest value of nn such that un>100u_n > 100. [2 marks]

Solutions

15MasterySAQ-SDefinition and general term of arithmetic sequencess10 marksPaper 2~15 min
A concert hall has seats arranged in rows. The number of seats in each row forms an arithmetic sequence. The first row has 2020 seats and the sixth row has 4040 seats.
(a)
Find the common difference of the sequence. [2 marks]
(b)
Calculate the number of seats in the fifteenth row. [2 marks]
(c)
The total number of seats in the first nn rows is 800800. Find the value of nn. [2 marks]
(d)
The concert hall has exactly 2525 rows. A second concert hall also has 2525 rows of seats. Its first row has 3030 seats and its last row has 7878 seats, with the number of seats in each row also forming an arithmetic sequence. Determine which concert hall has more total seats, showing your reasoning clearly. [4 marks]

Solutions

16MasterySAQ-SDefinition and general term of arithmetic sequencess8 marksPaper 2~12 min
A small business sells handmade candles. In the first week of operation, they sell 20 candles. Each subsequent week, sales increase by a constant number of candles. In the 1212th week, they sell 8686 candles.
(a)
Determine the weekly increase in sales. [2 marks]
(b)
Calculate the week number in which weekly sales will first exceed 150150 candles. [2 marks]
(c)
The business needs total cumulative sales to exceed 20002000 candles before it can break even. Determine the week in which the business first breaks even, and calculate the total sales in that week. [4 marks]

Solutions

17MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A construction company builds walls using bricks. The number of bricks in each layer forms an arithmetic sequence. The first layer has 5050 bricks and the seventh layer has 8080 bricks.
(a)
Find the common difference of the sequence. [2 marks]
(b)
Calculate the number of bricks in the fifteenth layer. [2 marks]
(c)
The total number of bricks used in the first nn layers is 300300. Find the value of nn. [2 marks]

Solutions

18MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A new fitness app tracks a user's daily step count. On the first day of a challenge, a user records 50005000 steps. Each subsequent day, the user increases their daily step count by a constant amount compared to the previous day. On the 1212th day of the challenge, the user records 83008300 steps.
(a)
Show that the common difference, dd, of this arithmetic sequence is 300300 steps per day. [2 marks]
(b)
Find the day number on which the user's step count first exceeds 12 00012\,000 steps. [2 marks]
(c)
The challenge lasts 3030 days. The app awards a bonus if the user's total step count over the full 3030 days exceeds 300 000300\,000 steps. Determine, with justification, whether the user earns the bonus. [2 marks]

Solutions

19MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
An athlete is training for a marathon. On the first day of training, she runs 5 km5\,\text{km}. Each subsequent day, she increases the distance she runs by a constant amount, d kmd\,\text{km}. After 20 days of training, the total distance she has run is 290 km290\,\text{km}.
(a)
Calculate the value of dd. [3 marks]
(b)
Find the minimum number of days of training required for her total distance to exceed 500 km500\,\text{km}. [3 marks]

Solutions

20MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A student saves money over 12 weeks. She saves USD 200 in the first week and increases her savings by a fixed amount dd each week. After 12 weeks her total savings are USD 4200.
(a)
Write down the value of u1u_1 and state what S12S_{12} represents in this context. [1 mark]
(b)
Calculate the common difference dd of the arithmetic sequence representing her weekly savings. [2 marks]
(c)
The student claims that by week 15 her weekly savings will exceed USD 400. Determine whether this claim is correct, justifying your answer. [3 marks]

Solutions

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