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

1MasterySAQ-SDefinition and general term of arithmetic sequencess7 marksPaper 2~11 min
A company is developing a new recycling program. In the first month, 1200 kg1200\,\text{kg} of waste are collected. The monthly amounts collected form an arithmetic sequence. The company collects a total of 6000 kg6000\,\text{kg} over the first four months.
(a)
Calculate the common difference of the sequence. [3 marks]
(b)
Calculate the total waste collected over the first three months. [2 marks]
(c)
Determine the month in which the amount of waste collected first exceeds 2500 kg2500\,\text{kg}. [2 marks]
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2MasterySAQ-SDefinition and general term of arithmetic sequencess7 marksPaper 2~11 min
A biologist studies the population of a fish species in a lake. The population in the first year of observation is 45004500 fish. The population decreases each year by a constant amount, forming an arithmetic sequence. In the sixth year, the population is 32003200 fish.
(a)
Calculate the common difference of the sequence. [2 marks]
(b)
Determine the population in the tenth year. [2 marks]
(c)
The biologist estimates that the population will fall below 10001000 fish in year kk. Calculate the value of kk. [3 marks]
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Solutions

3MasterySAQ-SDefinition and general term of arithmetic sequencess10 marksPaper 2~15 min

Data

un=u1+(n−1)dSn=n2(2u1+(n−1)d),n∈Z+u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(2u_1 + (n-1)d), \quad n \in \mathbb{Z}^+
A construction company is building a series of houses along a straight road. The first house is located 20 m20\,\text{m} from the start of the road. Each subsequent house is placed 12 m12\,\text{m} further from the start of the road than the previous house.
(a)
Write down the values of u1u_1 and dd for this arithmetic sequence. [2 marks]
(b)
Calculate the distance from the start of the road to the 3030th house. [2 marks]
(c)
The local council requires that no house may be placed more than 400 m400\,\text{m} from the start of the road. Determine the maximum number of houses that can be built. [3 marks]
(d)
The company also requires that the total distance covered by all houses (i.e. the sum of all individual house distances from the start of the road) must not exceed 8000 m8000\,\text{m}. Determine whether the constraint in part (c) or this new constraint is more restrictive, justifying your answer. [3 marks]
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4MasterySAQ-SSum of an 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 3 cm3\,\text{cm} taller than the previous step. The total height of the staircase is 1050 cm1050\,\text{cm}.
(a)
Calculate the number of steps in the staircase. [3 marks]
(b)
Building regulations require that no single step in a staircase may exceed 80 cm80\,\text{cm} in height. Determine whether this staircase complies with the regulation. [3 marks]
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5MasterySAQ-SSum of an arithmetic sequences6 marksPaper 2~9 min

Data

$ $
A construction company is building a straight section of road. On the first day, they lay 1212 metres of road. Each subsequent day, they lay 33 metres more than the previous day. The project requires a total of 12181218 metres of road to be laid.
(a)
Show that the number of days, nn, required to complete the road satisfies the equation $$3n^2 + 21n - 2436 = 0. \quad [2 marks]
(b)
Hence, calculate the number of days required to complete the road. $ [2 marks]
(c)
Determine the length of road laid on the final day. $ [2 marks]
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6MasterySAQ-SSum of an arithmetic sequences6 marksPaper 2~9 min
A farmer plants trees in rows. The first row contains 2020 trees. Each subsequent row contains 55 more trees than the previous row. The total number of trees planted is 14301430.
(a)
Show that the number of rows planted is 2222. [3 marks]
(b)
The farmer wishes to plant at least 20002000 trees using the same pattern (first row of 2020 trees, each subsequent row increasing by 55). Determine the minimum number of complete rows required. [3 marks]
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7MasterySAQ-SGeometric sequences and their general terms6 marksPaper 2~9 min
A construction company is building a series of retaining walls. The first wall requires 12001200 bricks. Each subsequent wall requires 4%4\% fewer bricks than the previous wall.
(a)
Calculate the number of bricks required for the 88th wall, giving your answer to the nearest whole brick. [2 marks]
(b)
Calculate the total number of bricks required for the first 1212 walls, giving your answer to the nearest whole brick. [2 marks]
(c)
The company has a total stock of 15 00015\,000 bricks. Determine the maximum number of complete walls they can build, and find the number of bricks remaining in stock after those walls are built. [2 marks]
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8MasterySAQ-SGeometric sequences and their general terms6 marksPaper 2~9 min

Data

un=u1rn−1u_n = u_1 r^{n-1}, Sn=u1(rn−1)r−1\quad S_n = \dfrac{u_1(r^n - 1)}{r - 1}, r≠1r \neq 1
A geometric sequence u1,u2,u3,…u_1, u_2, u_3, \ldots has first term u1=12u_1 = 12 and common ratio r=32r = \dfrac{3}{2}.
(a)
State the value of u4u_4. [1 mark]
(b)
Calculate the sum of the first 8 terms. Give your answer as an exact fraction. [2 marks]
(c)
Show that the sum of the first kk terms can be written in the form Sk=32k ⁣(3k−2k)⋅AS_k = \dfrac{3}{2^k}\!\left(3^k - 2^k\right) \cdot A for some constant AA, and hence determine the value of kk for which Sk=38 ⁣(3k−2k)S_k = \dfrac{3}{8}\!\left(3^k - 2^k\right). [3 marks]

Solutions

9MasterySAQ-SGeometric sequences and their general terms8 marksPaper 2~12 min
A biologist is studying the spread of a plant species in a lake. The area covered by the plants on the first day is 5 m25\,\text{m}^2. Each day, the area covered increases by 12%12\% of the previous day's area.
(a)
Calculate the area covered on the 15th day. Give your answer correct to 3 significant figures. [2 marks]
(b)
Calculate the total area covered from day 1 to day 20 inclusive. Give your answer correct to 3 significant figures. [2 marks]
(c)
The lake has a total surface area of 2500 m22500\,\text{m}^2. Determine the day on which the plants first cover more than 50%50\% of the lake's surface area. [2 marks]
(d)
Using your answer to part (c), determine the day on which the model predicts the plants would first cover the entire lake. Hence comment on whether the geometric model remains valid beyond the day found in part (c). [2 marks]
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10MasterySAQ-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

11MasterySAQ-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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12MasterySAQ-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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13MasterySAQ-SDefinition and general term of arithmetic sequencess8 marksPaper 2~12 min
A theatre has seats arranged in rows. The first row has 1515 seats. Each subsequent row has 44 more seats than the previous row. The last row has 9999 seats.
(a)
Calculate the number of rows in theatre. [3 marks]
(b)
The theatre charges USD 12 per seat. A production company rents the entire theatre for a performance. Calculate the maximum possible revenue from ticket sales. [2 marks]
(c)
The production company sells all seats except those in the last three rows. Determine the revenue lost compared to a full house. [3 marks]

Solutions

14MasterySAQ-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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15MasterySAQ-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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16MasterySAQ-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

17MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
The number of books on the shelves of a library forms an arithmetic sequence. The first shelf has 1818 books and the fourth shelf has 3030 books.
(a)
Find the common difference of the sequence. [2 marks]
(b)
Calculate the number of books on the twelfth shelf. [2 marks]
(c)
The total number of books on the first nn shelves is 360360. Find the value of nn, justifying why only one value is valid. [2 marks]

Solutions

18MasterySAQ-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

19MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min
A cyclist trains for a race. On the first day, she cycles 15 km15\,\text{km}. Each subsequent day, she increases the distance cycled by a constant amount compared to the previous day. On the 1010th day, she cycles 42 km42\,\text{km}.
(a)
Show that the daily increase is 3 km3\,\text{km}. [2 marks]
(b)
Calculate the total distance cycled over the first 1414 days. [2 marks]
(c)
The cyclist's target is to cycle a total of at least 1000 km1000\,\text{km} during her training. Determine the minimum number of days of training required to meet this target. [2 marks]

Solutions

20MasterySAQ-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

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