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

1FoundationSAQ-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]
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Solutions

2FoundationSAQ-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]
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Solutions

3FoundationSAQ-SDefinition and general term of arithmetic sequences8 marksPaper 1~12 min
A gardener plants trees in a straight line. The first tree is planted 5 m5\,\text{m} from a fence. Each subsequent tree is planted 4 m4\,\text{m} further from the fence than the previous tree.
(a)
State the value of the first term u1u_1 and the common difference dd of the arithmetic sequence of distances. [1 mark]
(b)
Calculate the distance from the fence of the 1212th tree. [2 marks]
(c)
The last tree is 81 m81\,\text{m} from the fence. Find the total number of trees planted. [2 marks]
(d)
A path is to be laid from the fence to the last tree, passing each tree in order. The total length of path equals the sum of all tree distances from the fence. Determine whether the total length of path exceeds 850 m850\,\text{m}, justifying your answer. [3 marks]
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Solutions

4FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 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} greater than the previous step. The heights of the steps form an arithmetic sequence.
(a)
Write down the values of u1u_1 and dd for this sequence. [1 mark]
(b)
Calculate the height of the 8th step. [2 marks]
(c)
The top step has a height of 60 cm60\,\text{cm}. Find the total number of steps in the staircase. [2 marks]
(d)
Find the total combined height of all the steps in the staircase. [2 marks]
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Solutions

5FoundationSAQ-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

6FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A theme park is building a new roller coaster. The first car travels 120 m120\,\text{m} before the first drop. Each subsequent car travels 8 m8\,\text{m} less than the previous car before reaching the first drop.
(a)
State whether the distances travelled by the cars form an arithmetic sequence. Justify your answer. [2 marks]
(b)
Find the distance travelled by the 55th car before the first drop. [2 marks]
(c)
The theme park states that any car travelling 0 m0\,\text{m} or less cannot reach the first drop and must be removed. Find the maximum number of cars that can reach the first drop. [3 marks]
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Solutions

7FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A movie theatre has 1818 seats in the first row. Each subsequent row has 44 more seats than the previous row.
(a)
Write down the value of the first term u1u_1 and the common difference dd of the arithmetic sequence formed by the number of seats in each row. [1 mark]
(b)
Calculate the number of seats in the 1010th row. [2 marks]
(c)
The last row has 7474 seats. Find the total number of seats in theatre. [4 marks]
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Solutions

8FoundationSAQ-SSum of an arithmetic sequence7 marksPaper 1~11 min
A concert hall has 1515 rows of seats. The number of seats in each row forms an arithmetic sequence. The first row has 1212 seats and the last row has 5454 seats.
(a)
Find the common difference dd of the sequence. [2 marks]
(b)
Find the total number of seats in the concert hall. [2 marks]
(c)
The concert hall charges USD 45 per seat in the first 88 rows and USD 30 per seat in the remaining rows. Determine the total ticket revenue when every seat is sold. [3 marks]

Solutions

9FoundationSAQ-SSum of an arithmetic sequence8 marksPaper 1~12 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 student saves money each week. In the first week, she saves 1515 dollars. Each week she saves 44 dollars more than the previous week.
(a)
Find the amount she saves in the 8th week. [2 marks]
(b)
Calculate the total amount saved after 8 weeks. [3 marks]
(c)
Find the minimum number of weeks required for her total savings to exceed 500500 dollars. [3 marks]

Solutions

10FoundationSAQ-SSum of an arithmetic sequence8 marksPaper 1~12 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 small business records its monthly profit. In the first month, the profit is 50005000 dollars. The profit increases by a fixed amount each month. In the sixth month, the profit is 65006500 dollars.
(a)
Write down an equation, in terms of dd, for the profit in the sixth month. [1 mark]
(b)
Find the common difference dd of this arithmetic sequence. [2 marks]
(c)
Find the total profit over the first 12 months. [2 marks]
(d)
Determine the minimum number of months required for the total profit to exceed 200 000200\,000 dollars. [3 marks]
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Solutions

11FoundationSAQ-SSum of an arithmetic sequence7 marksPaper 1~11 min
A small business owner records the number of customers served each day. The number of customers served forms an arithmetic sequence. On the first day, 88 customers are served. On the tenth day, 3535 customers are served.
(a)
Calculate the common difference, dd, of the sequence. [2 marks]
(b)
Find the total number of customers served over the first 1515 days. [2 marks]
(c)
The business owner claims that the total number of customers served will exceed 20002000 within the first 3030 days. Determine whether this claim is correct, showing your reasoning. [3 marks]
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Solutions

12FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A student is saving money to buy a laptop. She saves 1515 dollars in week 1. Each week she saves 55 dollars more than the previous week. The amounts saved each week form an arithmetic sequence.
(a)
Calculate the amount she saves in week 6. [2 marks]
(b)
Determine the week number in which she first saves 5555 dollars in a single week. [2 marks]
(c)
The laptop costs USD 500\text{USD } 500. Calculate the minimum number of weeks she must save before her total savings are enough to buy the laptop. [3 marks]

Solutions

13FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A small business owner tracks monthly sales. In January (month 1), sales were USD 5000. Sales increase by a fixed amount each month. In June (month 6), sales were USD 7500.
(a)
State the common difference of this arithmetic sequence. [1 mark]
(b)
Find the sales in the 10th month. [2 marks]
(c)
The owner needs total cumulative sales to exceed USD 200 000 to qualify for a business grant. Determine the minimum number of complete months required for cumulative sales to exceed USD 200 000, and state the month in which the owner first qualifies. [4 marks]
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Solutions

14FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min
A library has 10001000 books in its first year. Each year, the number of books increases by a constant amount. After 44 years, there are 16001600 books.
(a)
Find the common difference of this arithmetic sequence. [2 marks]
(b)
Find the number of books in the 1010th year. [2 marks]
(c)
Determine the first year in which the number of books exceeds 40004000. [3 marks]

Solutions

15FoundationSAQ-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

16FoundationSAQ-SDefinition and general term of arithmetic sequences8 marksPaper 1~12 min
A cyclist trains by increasing her daily distance by a constant amount each day. On day 1 she cycles 12 km12\,\text{km}. On day 4 she cycles 21 km21\,\text{km}.
(a)
Calculate the common difference, dd, of this arithmetic sequence. [2 marks]
(b)
Calculate the distance she cycles on day 10. [2 marks]
(c)
Show that she first cycles 48 km48\,\text{km} on day 13. [1 mark]
(d)
The cyclist's training plan runs for 13 days. Her coach states that the total distance cycled over the 13 days must exceed 390 km390\,\text{km} for the plan to be considered successful. Determine whether the training plan is successful. [3 marks]

Solutions

17FoundationSAQ-SSum of an arithmetic sequence7 marksPaper 1~11 min
Elena is training for a charity run. She runs 3 km3\,\text{km} on her first day and increases the distance she runs each day by a constant amount. On her 10th day, she runs 12 km12\,\text{km}.
(a)
Find the common difference, dd, of this arithmetic sequence. [2 marks]
(b)
Find the total distance Elena runs over the first 10 days. [2 marks]
(c)
Elena's training programme lasts nn days. Find the minimum value of nn such that her total distance first exceeds 200 km200\,\text{km}. [3 marks]
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Solutions

18FoundationSAQ-SSum of an arithmetic sequence7 marksPaper 1~11 min
A small business sells handmade candles. The number of candles sold each week forms an arithmetic sequence. In the first week, 2020 candles are sold. In the fifth week, 4040 candles are sold.
(a)
Find the common difference of the sequence. [2 marks]
(b)
Find the total number of candles sold over the first 55 weeks. [2 marks]
(c)
The business needs to sell a total of at least 500500 candles to break even. Determine the minimum number of complete weeks required to reach this target, assuming the arithmetic sequence continues. [3 marks]

Solutions

19FoundationSAQ-SSum of an arithmetic sequence8 marksPaper 1~12 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), Sn=n2(u1+un)\quad S_n = \dfrac{n}{2}(u_1 + u_n)
A company sells a product. In week 1, they sell 200200 units. Each week after that, the number of units sold increases by a constant amount dd. In week 12, they sell 530530 units.
(a)
Write down an expression for unu_n, the number of units sold in week nn, in the form un=u1+(n−1)du_n = u_1 + (n-1)d, stating the value of u1u_1. [1 mark]
(b)
Calculate the weekly increase in sales, dd. [2 marks]
(c)
Find the total number of units sold over the first 12 weeks. [2 marks]
(d)
The company needs to sell a total of at least 10 00010\,000 units to break even. Determine the minimum number of complete weeks required to reach this target. [3 marks]

Solutions

20FoundationSAQ-SSum of an arithmetic sequence10 marksPaper 1~15 min
A construction company is building a staircase. The heights of the steps, measured in centimetres, form an arithmetic sequence. The first step has a height of 15 cm15\,\text{cm} and the sixth step has a height of 30 cm30\,\text{cm}.
(a)
Find the common difference of the sequence. [2 marks]
(b)
Find the height of the 1010th step. [2 marks]
(c)
Find the sum of the heights of the first 1010 steps. [2 marks]
(d)
The maximum total height of a staircase permitted by building regulations is 400 cm400\,\text{cm}. Determine the maximum number of complete steps the staircase can have. [4 marks]

Solutions

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