Sequences Patterns and Functions — Free IB MYP4 Mathematics (Standard) Questions

1QuestionRecognizing and Continuing Arithmetic SequencesConcept Practice

4 marks~6 minCriterion A

A recent university graduate accepts a job with a starting salary of USD 60000 per year. The job offer includes an annual salary increase of USD 3000 for the next 10 years.

a

Write an arithmetic sequence that models the graduate's salary for the first 5 years.

b

Using the arithmetic sequence formula, calculate the graduate's salary in the 10th year. Show your working. The formula for the nth term of an arithmetic sequence is given by

a_n = a_1 + (n - 1)d

, where

a_n

is the nth term,

a_1

is the first term,

n

is the term number, and

d

is the common difference.

c

Calculate the total amount of money the graduate will earn over the 10-year period. The formula for the sum of an arithmetic series is given by

S_n = \frac{n}{2}(a_1 + a_n)

, where

S_n

is the sum of the first n terms,

a_1

is the first term, and

a_n

is the nth term.

d

Discuss the strengths and limitations of using this arithmetic sequence as a model to predict the graduate's future earnings, considering factors such as inflation and potential promotions.

Solutions

2QuestionCreating a Rule for the nth TermConcept Practice

2 marks~3 minCriterion A

A gardener records the height of a bamboo plant over four consecutive days:

Day 1: 8 cm
Day 2: 11 cm
Day 3: 14 cm
Day 4: 17 cm

a

Determine the formula for

h_n

, the height of the plant (in cm) after

n

days. [1]

b

The gardener claims this linear model will reliably predict the plant's height after one year. Justify whether this claim is valid. [1]

Solutions

3QuestionIdentifying Domain and Range from TablesConcept Practice

2 marks~3 minCriterion B

A delivery drone records the following data during a test flight:

Time,

x

(minutes):

1, 2, 3, 4

Distance from base,

y

(metres):

3, 5, 7, 9

a

State the domain and range of the function represented by this data. [1]

b

The drone's safe operating limit is a maximum distance of 8 metres from base. Using the rule relating

x

to

y

, justify whether the drone exceeds this limit during the flight. [1]

Solutions

4QuestionApplications in Real-World GraphsConcept Practice

2 marks~3 minCriterion A

The graph below shows the height

h

(in metres) of a ball thrown vertically upward as a function of time

t

(in seconds). The ball is released from ground level, reaches a maximum height of

20

m at

t = 2

s, and returns to the ground at

t = 4

s.

a

State the domain and range of this function using interval notation. [1]

b

The ball must clear a barrier of height

15

m. Using the graph, identify the interval of time during which the ball is above

15

m, and justify whether the ball successfully clears the barrier. [1]

Solutions

5QuestionDeriving the nth Term of an Arithmetic SequenceAssessment Practice

6 marks~9 minCriterion C

A student records weekly savings (in dollars) over four weeks:

Week 1: 15 — Week 2: 22 — Week 3: 29 — Week 4: 36

A friend claims the

n

th term of this arithmetic sequence is

u_n = 8n + 7

.

a

Calculate the values predicted by

u_n = 8n + 7

for Weeks 1 to 4. [2]

b

Deduce, with supporting calculations, whether the formula

u_n = 8n + 7

correctly models the student's savings. [2]

c

The student has a savings goal of 100 dollars in a single week. Using the correct

n

th term formula, justify whether this goal will ever be reached exactly under the same saving pattern. [2]

Solutions

6QuestionSolving Problems Using Arithmetic SequencesAssessment Practice

20 marks~30 minCriterion B

Communication: Arithmetic Series and Staircase Patterns

A staircase is being built using identical blocks. The number of blocks required for each step forms an arithmetic sequence.

The pattern begins as follows:

Step number: 1, 2, 3, 4, …
Blocks in step: 3, 5, 7, 9, …

Let

S_n

represent the total number of blocks needed to build a staircase with

n

steps.

Question

a

Calculate

S_1

,

S_2

,

S_3

, and

S_4

. Show clearly how each total is formed from the blocks in each step. [4 marks]

b

Predict a general formula for

S_n

in terms of

n

. Your answer should:

- Identify the pattern in the totals,
- Express the relationship using algebra,
- Clearly define what

n

and

S_n

represent. [4 marks]

c

Justify your formula by showing that

S_n

can be written as the sum of the first

n

terms of an arithmetic series. In your response:

- Identify the first term

a

,
- Identify the common difference

d

,
- Write the

n

th term of the sequence,
- Use the arithmetic series sum formula,
- Simplify your expression to match your formula from part (b). [6 marks]

d

Derive the standard formula for the sum of an arithmetic series:

S_n = \frac{n}{2} [2a + (n-1)d]

. Use clear mathematical communication and explain each step of your derivation. [4 marks]

e

Explain why clear notation is important when solving this problem, especially the difference between:

- Blocks in the

n

th step,
- Total blocks in

n

steps. [2 marks]

Solutions

7QuestionFinding the Common DifferenceAssessment Practice

2 marks~3 minCriterion D

A student deposits money into a savings account each week. The account balance at the start of each week is recorded below.

Week	1	2	3	4	5
Balance (dollars)	10	15	20	25	30

a

Calculate the common difference in the account balance and explain what it represents in this savings context. [1]

b

The student wants to reach a balance of 60 dollars. State one assumption this arithmetic model makes, and justify whether the model is reliable for predicting when the target balance will be reached. [1]

Solutions

8QuestionEvaluating Composite FunctionsAssessment Practice

6 marks~9 minCriterion B

A computer simulation models the spread of a digital signal through a network. At each step, the signal passes through two filters in sequence: filter

f

doubles the strength, then filter

g

adds 1 unit. Formally,

f(x) = 2x

and

g(x) = x + 1

, giving composite function

h(x) = g(f(x))

. The simulation begins at strength

x_0 = 1

and updates by

x_{n+1} = h(x_n)

.

a

Calculate

x_1

,

x_2

,

x_3

, and

x_4

. [2]

b

Deduce a general formula for

x_n

in terms of

n

. [2]

c

The network has a maximum capacity of 1000 units. Justify whether this capacity is exceeded within the first 8 steps of the simulation. [2]

Solutions

9QuestionUnderstanding Composite Functions NotationAssessment Practice

6 marks~9 minCriterion D

A small business models its pricing using two functions:

C(x) = 5x + 20

, where

C

is the production cost in dollars for

x

units, and

P(c) = 1.15c

, which applies a 15% markup to any cost

c

.

a

Write the composite function

P(C(x))

, expanding your answer fully. [1]

b

Using your composite function, calculate the selling price for a batch of 50 units. Show all working. [2]

c

The business owner is deciding whether to use this model for batches ranging from 10 units to 500 units. Advise the owner whether the model should be applied uniformly across this range, justifying your advice with reference to at least one strength and one limitation of

P(C(x))

. [3]

Solutions

10QuestionEvaluating Composite FunctionsAssessment Practice

2 marks~3 minCriterion A

A surveyor uses a 3-4-5 right triangle to set a reference angle

\theta

, where

\theta

is the angle opposite the side of length 3.

The functions

f(x) = \sin(x°)

and

g(x) = x + 30

model how the surveyor adjusts bearing angles.

Calculate the value of

f(g(\theta))

, giving your answer to 3 significant figures. [2]

Solutions

11QuestionEvaluating Composite FunctionsAssessment Practice

2 marks~3 minCriterion C

A car rental company charges 50 dollars per day plus 0.20 dollars per kilometre driven. A sales representative drives a fixed route of 180 km each working day.

Let

d

= number of days rented and

k(d) = 180d

= total kilometres driven.

The total rental cost is modelled by

C(d) = 50d + 0.20 \times k(d)

.

a

Explain how

C(d)

represents a composite function in this context. [1]

b

Interpret one limitation of using

C(d)

to predict the representative's actual travel costs. [1]

Solutions

12QuestionEvaluating Terms Using the nth Term FormulaAssessment Practice

5 marks~8 minCriterion B

A community garden is being expanded each year. The total number of plant beds after year

n

is modelled by

a_1 = 1

and

a_{n+1} = a_n + 2n + 1

for

n \geq 1

.

a

Calculate the first four terms of the sequence, showing all working. [2]

b

Deduce a formula for the

n

th term

a_n

. [1]

c

Justify that your formula is valid for all integers

n \geq 1

, then advise the garden planners whether they should expect resource requirements to increase at a constant rate as the garden grows. [2]

Solutions

13QuestionDistinguishing Between Linear and Non-Linear SequencesAssessment Practice

5 marks~8 minCriterion C

A solar panel installation company charges a fixed connection fee plus a constant rate per panel installed. A technician records the following data from five recent jobs:

Number of panels ( $x$ )	1	2	3	4	5
Total cost in hundreds of dollars ( $y$ )	3	7	11	15	19

a

Justify whether the relationship between

x

and

y

is linear or non-linear. [2]

b

Derive a formula for

y

in terms of

x

. [1]

c

A client has a budget of 1800 dollars. Advise the client whether this budget is sufficient for an installation of 5 panels, and explain what the

y

-intercept represents in this context. [2]

Solutions

14QuestionFinding Term Position Given ValueAssessment Practice

7 marks~11 minCriterion D

A student opens a savings account with an initial deposit of 500 dollars. Each month, exactly 25 dollars is added, forming an arithmetic sequence.

a

Show that the account balance first reaches 875 dollars in month 16. [2]

b

The student considers making an additional one-off deposit of 200 dollars at the start of month 10. Deduce the new month in which the balance would first reach 875 dollars under this revised plan. [2]

c

The arithmetic model assumes a constant monthly increase of 25 dollars. Advise the student whether this model is reliable for predicting their long-term account balance, discussing at least two real-world factors in your response. [3]

Solutions

15QuestionUnderstanding Function Notation f of xAssessment Practice

6 marks~9 minCriterion B

A biologist models the growth of a bacterial colony. The population (in thousands) at each hourly stage is defined by

a_1 = 2

and the recurrence relation

a_{n+1} = f(a_n)

, where

f(x) = 3x - 1

.

The first four hourly populations are:

a_1 = 2

,

a_2 = 5

,

a_3 = 14

,

a_4 = 41

a

Calculate

f(f(f(2)))

. [2]

b

Deduce a general formula for

a_n

in terms of

n

. [3]

c

The laboratory has capacity for 400 thousand bacteria. Using your formula from part (b), justify whether the laboratory capacity is exceeded at stage 6. [1]

Solutions

16QuestionDefining Piecewise and Step Functions IntroductoryAssessment Practice

3 marks~5 minCriterion C

A postal company charges for packages according to the step function shown in the graph below.

The cost (in dollars) remains constant within each weight interval and jumps at specific weight boundaries.

a

Explain what the open and closed circles at each boundary point on the graph indicate about which interval includes that boundary weight. [1]

b

Deduce the cost of posting a package weighing 3.5 kg. [1]

c

A customer has a budget of 9 dollars. Advise the customer whether this budget is sufficient to post a package weighing exactly 4 kg, and what weight limit they must stay within to achieve a lower cost. [1]

Solutions

17QuestionInterpreting Function Values in ContextAssessment Practice

4 marks~6 minCriterion A

The height

h

(in metres) of a ball thrown upward is modelled by a downward-opening parabola with vertex

(2,\ 20)

, intersecting the

t

-axis at

t = 0

s and

t = 4

s, where

t

is time in seconds.

a

Interpret the graph to describe how the height of the ball changes over the four-second flight. [1]

b

Deduce the equation of the parabola in the form

h(t) = a(t - p)^2 + q

, and hence calculate the height of the ball at

t = 1

s. [2]

c

A safety sensor triggers if the ball remains above

15

m for longer than

1.5

seconds. Justify whether the sensor is triggered during the flight. [1]

Solutions

18QuestionInterpreting Function Values in ContextAssessment Practice

8 marks~12 minCriterion D

A city council models the total cost (in dollars) of running a community recycling program for

t

months using

C(t) = 5000 + 120t

, where 5000 dollars represents the initial setup cost. The total benefit (in dollars) from reduced landfill fees is modelled by

B(t) = 2000\sqrt{t}

.

a

Calculate

C(12)

and

B(12)

. Interpret both results in the context of the recycling program. [2]

b

Justify whether the recycling program is financially worthwhile after 12 months. [2]

c

Advise the city council on whether these two functions are sufficient to support long-term financial decision-making. In your response, identify at least two assumptions made by each function and at least two limitations of the models overall. [4]

Solutions

19QuestionIdentifying Discrete and Continuous FunctionsAssessment Practice

4 marks~6 minCriterion C

A museum records two data sets over six hours: the number of visitors counted each hour (whole numbers only) and the water temperature in an aquarium tank measured continuously.

Visitor counts (hours 1–6): 15, 22, 18, 27, 20, 24.
Water temperature (°C): recorded at every instant throughout the six hours.

a

State which data set represents a discrete function and which represents a continuous function. [1]

b

Calculate the range of the visitor counts over the six hours. [1]

c

The museum manager claims the aquarium temperature data is more useful than the visitor counts for detecting sudden environmental changes. Justify this claim using the mathematical properties of continuous and discrete functions. [2]

Solutions

20QuestionIdentifying Domain and Range from TablesAssessment Practice

6 marks~9 minCriterion D

A wildlife park has a daily water supply of 5000 litres. Its largest enclosure (Enclosure A) must receive at least 200 litres per day. Daily water consumption (in litres) for three enclosures over one week is recorded below.

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Enclosure A (L)	1200	1300	1250	1400	1350	1500	1450
Enclosure B (L)	800	750	900	850	700	950	1000
Enclosure C (L)	600	650	550	700	600	750	800

a

State the domain and calculate the range of the function representing total daily water consumption across all three enclosures. [2]

b

Deduce whether the park's weekly schedule is feasible under both the 5000-litre supply constraint and the 200-litre minimum for Enclosure A. Support your answer with calculations. [2]

c

Justify whether the constant daily-value model provides a reliable basis for the park's water management decisions, identifying two real-world factors that affect your judgement. [2]

Solutions

Sequences Patterns and Functions — Free MYP4 Mathematics (Standard) Practice Questions