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Arithmetic Sequences and Series · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min

Data

The total distance from the first tree to the fifth tree is the sum of the four consecutive gaps.
A farmer is planting trees in a straight line. The distance from the first tree to the second tree is 5 m5\,\text{m}. The distance between consecutive trees increases by a constant amount, forming an arithmetic sequence. The total distance from the first tree to the fifth tree is 38 m38\,\text{m}.
(a)
State the value of u1u_1, the first gap, and the value of S4S_4, the total of the first four gaps. [2 marks]
(b)
Calculate the common difference, dd, of the arithmetic sequence of gaps. [2 marks]
(c)
Hence, find the total distance from the first tree to the tenth tree. [1 mark]
diagram

Solutions

2FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min
A company is stacking boxes in a warehouse. The bottom row has 3030 boxes. Each row above has 44 fewer boxes than the row below, forming an arithmetic sequence. The top row has 66 boxes.
(a)
State the value of u1u_1 and the common difference dd. [1 mark]
(b)
Calculate the number of rows of boxes. [2 marks]
(c)
Find the total number of boxes in the stack. [2 marks]
diagram

Solutions

3FoundationSAQ-SSum of an arithmetic sequence5 marksPaper 1~8 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)
In a small business, the monthly sales of a product form an arithmetic sequence. The sales in the first month were USD 1200, and the sales in the third month were USD 1500.
(a)
State the common difference of the sequence. [1 mark]
(b)
Calculate the total sales over the first 1010 months. [2 marks]
(c)
The business owner claims that monthly sales will exceed USD 3000 for the first time during the second year of trading (months 13–24). Determine whether this claim is correct, justifying your answer. [2 marks]
diagram

Solutions

4FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min
A small business owner tracks the monthly profit of her company. In month 1, the profit was USD 2000. The profit increases by a fixed amount each month, forming an arithmetic sequence. The profit in month 10 was USD 3800.
(a)
Write down an equation, in terms of dd, for the profit in month 10. [1 mark]
(b)
Calculate the common difference, dd, of the sequence. [2 marks]
(c)
Hence, find the profit in month 15. [2 marks]

Solutions

5FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min
A theatre has 2525 rows of seats. The number of seats in each row forms an arithmetic sequence. The first row has 1212 seats and the second row has 1515 seats.
(a)
Write down an expression for unu_n, the number of seats in the nn-th row. [1 mark]
(b)
Calculate the number of seats in the 1010th row. [2 marks]
(c)
The theatre management claims that one row contains exactly 4848 seats. Determine which row this is, and verify that this row exists within theatre. [2 marks]

Solutions

6FoundationSAQ-SSum of an arithmetic sequence5 marksPaper 1~8 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)
A concert hall has 2020 rows of seats. The number of seats in each row forms an arithmetic sequence. The first row has 3030 seats and the second row has 3434 seats.
(a)
Write down the common difference and the first term of the sequence. [1 mark]
(b)
Calculate the number of seats in the 1515th row. [2 marks]
(c)
The concert hall is extended by adding further rows, each following the same arithmetic sequence. The venue manager requires a total capacity of at least 25002500 seats across all rows. Determine the minimum number of rows needed to meet this requirement. [2 marks]

Solutions

7FoundationSAQ-SSum of an arithmetic sequence5 marksPaper 1~8 min
A gardener plants trees in a straight line. The distance from the first tree to the second tree is 2.5 m2.5\,\text{m}, and each subsequent gap between consecutive trees is 1.2 m1.2\,\text{m} more than the previous gap. The gaps form an arithmetic sequence.
(a)
State the first term u1u_1 and common difference dd of this arithmetic sequence. [1 mark]
(b)
Find the value of nn such that the gap between the nnth and (n+1)(n+1)th trees first exceeds 15 m15\,\text{m}. [2 marks]
(c)
The gardener has a plot of length 200 m200\,\text{m}. Determine the maximum number of trees that can be planted in a single straight line within this plot. [2 marks]
diagram

Solutions

8FoundationSAQ-SApplications of arithmetic sequences in real-world contexts5 marksPaper 1~8 min
A water tank contains 200200 litres of water at the start of day 1. Each day, 88 litres of water are added to the tank.
(a)
State the value of the first term, u1u_1, and the common difference, dd, for the arithmetic sequence representing the volume of water in the tank at the start of each day. [1 mark]
(b)
Calculate the volume of water in the tank at the start of the 1414th day. [2 marks]
(c)
The tank has a maximum capacity of 500500 litres. Determine the day on which the volume of water in the tank first reaches or exceeds 500500 litres, and state the volume on that day. [2 marks]

Solutions

9FoundationSAQ-SApplications of arithmetic sequences in real-world contexts5 marksPaper 1~8 min
Mia starts a new job with annual salary of 32 00032\,000 dollars. She receives a fixed annual pay rise of 1 5001\,500 dollars each year at the end of each year of employment.
(a)
State the value of the first term, u1u_1, and the common difference, dd, for the arithmetic sequence representing her annual salary. [1 mark]
(b)
Calculate her annual salary in her 88th year of employment. [2 marks]
(c)
Determine the first year of employment in which her total cumulative earnings exceed 500 000500\,000 dollars. [2 marks]
diagram

Solutions

10FoundationSAQ-SApplications of arithmetic sequences in real-world contexts7 marksPaper 1~11 min
A farmer plants trees in rows. The first row contains 1515 trees. Each subsequent row contains 44 more trees than the previous row.
(a)
Calculate the number of trees in the 1212th row. [2 marks]
(b)
Determine the first row number in which the number of trees exceeds 7575. [2 marks]
(c)
The farmer has space for exactly 2020 rows but a total budget allowing for at most 900900 trees. Determine whether the farmer can plant trees in all 2020 rows without exceeding the budget. Justify your answer. [3 marks]

Solutions

11FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min
A cyclist trains for a race by increasing her daily cycling distance by a fixed number of kilometres each day. On day 1 she cycles 8 km8\,\text{km}, and on day 7 she cycles 26 km26\,\text{km}.
(a)
Calculate the common difference, dd, of the arithmetic sequence. [2 marks]
(b)
Find on which day she first cycles more than 50 km50\,\text{km}. [2 marks]
(c)
The cyclist's training plan lasts 20 days. Determine the total distance she cycles over the full 20-day period. [1 mark]

Solutions

12FoundationSAQ-SApplications of arithmetic sequences in real-world contexts5 marksPaper 1~8 min
A concert hall has 2525 seats in the first row. Each row behind the first has 66 more seats than the row in front of it.
(a)
State the value of the first term, u1u_1, and the common difference, dd, for the arithmetic sequence representing the number of seats per row. [1 mark]
(b)
Calculate the number of seats in the 1010th row. [2 marks]
(c)
The hall has nn complete rows. Given that the total number of seats must exceed 20002000, find the minimum value of nn. [2 marks]

Solutions

13FoundationSAQ-SSum of an arithmetic sequence6 marksPaper 1~9 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)
A runner is training for a race. On the first day, she runs 3 km3\,\text{km}. Each subsequent day, she runs 0.8 km0.8\,\text{km} more than the previous day. The distances she runs each day form an arithmetic sequence.
(a)
State the first term and the common difference of the sequence. [1 mark]
(b)
Determine the first day on which she runs more than 10 km10\,\text{km}. [2 marks]
(c)
Find the minimum number of complete days of training required for her total distance to exceed 200 km200\,\text{km}. [3 marks]

Solutions

14FoundationSAQ-SSum of an arithmetic sequence6 marksPaper 1~9 min
A construction company is building a staircase. The first step has a height of 15 cm15\,\text{cm}, and each subsequent step is 4 cm4\,\text{cm} higher than the previous one. The heights of the steps form an arithmetic sequence.
(a)
State the first term and the common difference of the sequence. [1 mark]
(b)
Determine the step number that first has a height exceeding 50 cm50\,\text{cm}. [2 marks]
(c)
The total height from the ground to the top of the nnth step must not exceed 700 cm700\,\text{cm}. Determine the maximum number of complete steps that can be built. [3 marks]

Solutions

15FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A construction company is building a new housing estate. The first house is built 20 m20\,\text{m} from the main road. Each subsequent house is built 12 m12\,\text{m} further from the road than the previous house.
(a)
State the value of u1u_1 and the value of dd for this arithmetic sequence. [2 marks]
(b)
Find the distance from the main road to the 1515th house. [2 marks]
(c)
The estate has nn houses in total. The furthest house must be built no more than 500 m500\,\text{m} from the main road. Find the maximum value of nn. [3 marks]
diagram

Solutions

16FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A gardener plants trees in a straight line. The first tree is planted 5 m5\,\text{m} from the garden gate. Each subsequent tree is planted 3 m3\,\text{m} further from the gate than the previous tree.
(a)
Find the distance from the gate to the 1010th tree. [2 marks]
(b)
The distance from the gate to the kkth tree is 53 m53\,\text{m}. Find the value of kk. [2 marks]
(c)
The gardener has a section of garden that extends 200 m200\,\text{m} from the gate. Determine the maximum number of trees that can be planted within this section, and find the total distance covered from the gate to the last tree planted. [3 marks]

Solutions

17FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min

Data

un=u1+(n−1)dSn=n2(u1+un)u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(u_1 + u_n)
A cinema has 3030 seats in the first row. Each subsequent row has 44 more seats than the previous row. The last row has 126126 seats.
(a)
State the value of u1u_1 and the common difference dd. [1 mark]
(b)
Find the number of the row that has 126126 seats. [2 marks]
(c)
Find the total number of seats in the cinema. [2 marks]

Solutions

18FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A scientist monitors the growth of a bacterial colony. On day 1 there are 500500 bacteria. The number of bacteria increases by the same amount each day. On day 8 there are 14801480 bacteria.
(a)
Write down the general term unu_n for the number of bacteria on day nn, identifying the values of u1u_1 and dd. [1 mark]
(b)
Show that the common difference d=140d = 140. [2 marks]
(c)
Find the number of bacteria on day 15. [2 marks]
(d)
The scientist claims the colony will exceed 50005000 bacteria before day 35. Determine whether this claim is correct. [2 marks]
diagram

Solutions

19FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A small business owner deposits USD 2000 into a new savings account on January 1st, 2023. Each subsequent year, on January 1st, they deposit USD 150 more than the previous year's deposit. The deposits form an arithmetic sequence.
(a)
Write down the values of u1u_1 and dd for this arithmetic sequence. [1 mark]
(b)
Calculate the amount deposited on January 1st, 2030. [2 marks]
(c)
Calculate the total amount deposited from January 1st, 2023 to January 1st, 2030 inclusive. [2 marks]
(d)
Determine the first year in which the annual deposit exceeds USD 3500. [2 marks]

Solutions

20FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min
A scientist records the temperature of a chemical solution every minute. The temperature at the start (minute 0) is 25∘C25^\circ\text{C}. The temperature increases by a constant amount each minute, forming an arithmetic sequence. After 8 minutes, the temperature is 49∘C49^\circ\text{C}.
(a)
Find the common difference of the sequence. [3 marks]
(b)
Find the sum of all temperatures recorded from minute 0 to minute 12 inclusive. [2 marks]

Solutions

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