Free IB DP Practice Questions | RevisionPrep

1MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

The first term of an arithmetic sequence is

u_1 = 12

. The fifth term is

u_5 = 28

.

(a)

State the common difference,

d

. [1 mark]

(b)

Determine the general term

u_n

in the form

u_n = an + b

, where

a, b \in \mathbb{Z}

. [2 marks]

(c)

Find the smallest value of

n

for which the sum of the first

n

terms exceeds

1000

. [3 marks]

Solutions

2MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A company produces wooden pallets. In week 1, they produce 500 pallets. The production increases by a fixed number each week, forming an arithmetic sequence. In week 10, they produce 680 pallets.

(a)

State the weekly increase in production. [1 mark]

(b)

Determine an expression for the number of pallets produced in week

n

, denoted

P_n

. [2 marks]

(c)

The company receives a contract requiring a cumulative total of at least 40 000 pallets. Determine the minimum number of complete weeks needed to fulfil this contract. [3 marks]

Solutions

3MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

A construction company is building a terraced garden. The first terrace requires

120

bricks. Each subsequent terrace requires

8

fewer bricks than the previous one. Terraces are built in sequence for as long as the number of bricks required is positive. Formulae:

u_n = u_1 + (n-1)d

;

\quad S_n = \dfrac{n}{2}(2u_1 + (n-1)d)

(a)

Calculate the number of bricks required for the

10

th terrace. [2 marks]

(b)

Determine the total number of terraces that can be built, and hence calculate the total number of bricks used. [4 marks]

Solutions

4MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

Data

u_n = u_1 + (n-1)d

;

\quad S_n = \dfrac{n}{2}(2u_1 + (n-1)d)

A farmer plants trees in rows. The first row has

25

trees. Each subsequent row has a constant number of additional trees. The

12

th row has

69

trees.

(a)

Calculate the common difference of the arithmetic sequence. [2 marks]

(b)

Calculate the total number of trees planted in the first

20

rows. [3 marks]

(c)

The farmer has space for at most

30

rows and a total budget allowing a maximum of

3000

trees. Determine which constraint is reached first, justifying your answer. [1 mark]

Solutions

5MasterySAQ-SSum of an arithmetic sequence6 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.

(a)

Calculate the number of seats in the

12

th row. [2 marks]

(b)

Determine the total number of seats in the cinema. [2 marks]

(c)

The cinema plans to add a new section. The new section begins immediately after the existing 25 rows and continues the same arithmetic pattern. The new section must contain at least 1000 seats. Determine the minimum number of additional rows needed, and state the total number of seats in the new section with this minimum number of rows. [2 marks]

Solutions

6MasterySAQ-SApplications of arithmetic sequences in real-world contexts6 marksPaper 2~9 min

Data

u_n = u_1 + (n-1)d, \qquad S_n = \frac{n}{2}(2u_1 + (n-1)d)

A small business owner installs a solar panel system to reduce electricity costs. The system generates electricity at a rate that increases linearly each month as more panels are activated. In the first month, the system generates

1200

kWh. In the fifth month, it generates

2400

kWh. The owner assumes this linear trend continues.

(a)

Calculate the common difference of the arithmetic sequence representing the monthly generation. [2 marks]

(b)

Determine the total electricity generated over the first

12

months. [2 marks]

(c)

The owner's target is to generate a cumulative total of at least

50\,000

kWh. Determine the minimum number of complete months required to meet this target. [2 marks]

Solutions

7MasterySAQ-SApplications of arithmetic sequences in real-world contexts6 marksPaper 2~9 min

A company drills a well for geothermal energy. The drilling cost for the first metre is

800

dollars. The cost for each subsequent metre increases by a fixed amount. The cost for the

10

th metre is

1520

dollars.

(a)

Calculate the common difference of the arithmetic sequence representing the cost per metre. [2 marks]

(b)

Determine the total cost to drill the first

20

metres. [2 marks]

(c)

The company has a budget of

30\,000

dollars. Calculate the maximum number of complete metres that can be drilled without exceeding the budget. [2 marks]

Solutions

8MasterySAQ-SDefinition and general term of arithmetic sequences8 marksPaper 2~12 min

A biologist records the height of a bamboo plant each day. On day

1

, the height is

15\,\text{cm}

. The height increases by a constant daily amount, forming an arithmetic sequence. On day

12

, the height is

81\,\text{cm}

.

(a)

Show that the daily growth rate is

d = 6\,\text{cm per day}

. [1 mark]

(b)

Determine an expression for the height

H_n\,(\text{cm})

on day

n

. [2 marks]

(c)

Calculate the day on which the height first exceeds

200\,\text{cm}

. [2 marks]

(d)

The biologist claims that, at this growth rate, the total height gained over the first

40

days will exceed

4\,\text{m}

. Determine whether this claim is correct, showing all reasoning. [3 marks]

Solutions

9MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A small business sells handmade candles. In the first month, they sell 60 candles. Their sales increase by a fixed number each month. In the

6

th month, they sell

110

candles. The number of candles sold in month

n

is

c_n

.

(a)

Show that the monthly increase in sales is

10

candles per month. [2 marks]

(b)

Calculate the number of candles sold in month

12

. [2 marks]

(c)

The business needs to sell a total of at least

3000

candles to break even. Determine the minimum number of complete months required to reach this total. [2 marks]

Solutions

10MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A library has a membership drive. In the first month,

60

new members join. The library expects the number of new members each month to increase by a constant amount

d

.

(a)

In the

7

th month,

108

new members join. Calculate

d

. [2 marks]

(b)

Calculate the number of new members predicted to join the

18

th month. [2 marks]

(c)

The library's goal is to reach a cumulative total of at least

3000

new members. Determine the first month by the end of which this goal is met. [2 marks]

Solutions

11MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

An environmental scientist is tracking the growth of a bamboo plant. The height of the plant on day 1 is

40\,\text{cm}

. It grows at a constant rate each day. On day 12, the height is

106\,\text{cm}

. The height on day

n

is denoted by

h_n

.

(a)

Calculate the common difference

d

, the daily growth in cm. [2 marks]

(b)

Calculate the height of the plant on day 30. [2 marks]

(c)

Determine the day on which the height first reaches at least

200\,\text{cm}

. [2 marks]

Solutions

12MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A training program requires an athlete to run a certain distance each week. In week 1, she runs

5\,\text{km}

. Each week she increases the distance by a constant amount. In week 10, she runs

23\,\text{km}

. The distance run in week

n

is

d_n\,\text{km}

.

(a)

Calculate the weekly increase in distance. [2 marks]

(b)

Calculate the distance she will run in week 25. [2 marks]

(c)

The athlete's training program lasts

N

complete weeks. She must run a total of at least

500\,\text{km}

over the program. Determine the minimum value of

N

. [2 marks]

Solutions

13MasterySAQ-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\,\text{cm}

. Each subsequent step is

2\,\text{cm}

higher than the previous step.

(a)

Show that the heights of the steps form an arithmetic sequence with common difference

d = 2\,\text{cm}

. [1 mark]

(b)

Calculate the height of the

15

th step. [2 marks]

(c)

The total combined height of all steps must equal exactly

\text{620}\,\text{cm}

. Determine the number of steps in the staircase. [3 marks]

Solutions

14MasterySAQ-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

24

blocks. Each subsequent layer contains

3

fewer blocks than the layer below it. The top layer contains exactly

3

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\,\text{kg}

. The construction company can transport a maximum of

800\,\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]

Solutions

15MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A cinema has

18

seats in the first row. Each subsequent row has

4

more seats than the row in front of it. The last row has

82

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

16MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A movie theatre has 20 rows of seats. The number of seats in each row forms an arithmetic sequence. The first row has 15 seats and the second row has 18 seats.

(a)

State the common difference of the sequence. [1 mark]

(b)

Calculate the number of seats in the

15

th row. [2 marks]

(c)

The theatre manager claims the venue can seat more than 1000 people in total across all 20 rows. Determine whether this claim is correct. [3 marks]

Solutions

17MasterySAQ-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\,\text{cm}

. Each subsequent step has a height that is

3\,\text{cm}

more than the previous step. The staircase has

20

steps.

(a)

Show that the height of the

n

th step is given by

u_n = 3n + 9\,\text{cm}

. [2 marks]

(b)

Calculate the height of the

15

th step. [2 marks]

(c)

The total height of the staircase is the sum of the heights of all

20

steps. A building regulation states that the total height of a staircase must not exceed

4.5\,\text{m}

. Determine whether this staircase satisfies the regulation. [4 marks]

Solutions

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

d

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]

Solutions

19MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

A farmer plants trees in rows. The first row has

10

trees. Each subsequent row has

3

more trees than the previous row. The total number of trees planted is

770

.

(a)

Determine the number of rows planted. [4 marks]

(b)

The farmer has space for at most

75

trees in any single row. Given your answer to part (a), determine the number of trees in the last row and hence state whether the farmer's land constraint is satisfied. Justify your answer. [2 marks]

Solutions

20MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

A small business sells handmade candles. In week 1, they sell

50

candles. Each week they sell a fixed number more than the previous week. In week 7, they sell

86

candles.

(a)

Determine the weekly increase in sales. [2 marks]

(b)

Calculate the total number of candles sold in the first

12

weeks. [2 marks]

(c)

The business has a storage cost of USD 0.40 per unsold candle per week. They produce exactly

90

candles every week regardless of sales. Determine the week in which cumulative production first exceeds cumulative sales, and calculate the total storage cost incurred in that week. [2 marks]

Solutions

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