Sequences Patterns and Functions — Free IB MYP4 Mathematics (Extended) 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

2QuestionFinding Term Position Given ValueConcept Practice

2 marks~3 minCriterion C

A lighting technician programs a theatre spotlight to increase in brightness by a fixed amount each second. The brightness at second 1 is 12 lumens, and the brightness increases by 8 lumens per second.

Deduce the position

n

of the term in this arithmetic sequence that has value

a_n

, expressing

n

in terms of

a_n

,

a_1

, and

d

. Show all algebraic steps clearly. [2]

Solutions

3QuestionUsing Function Machines and DiagramsConcept Practice

2 marks~3 minCriterion A

A delivery company charges customers based on parcel weight. A function machine models the pricing structure:

Weight (kg)	3	5	7
Charge (dollars)	11	19	27

a

Deduce the rule for this function machine, expressing your answer using correct function notation. [1]

b

The company states: "No parcel costing more than 35 dollars will be accepted." A customer has a parcel weighing 9 kg. Justify whether the company will accept this parcel. [1]

Solutions

4QuestionWriting Inequality Notation for Domain and RangeConcept Practice

2 marks~3 minCriterion B

A physiotherapist records a patient's pain score (on a scale of 1 to 11) during five consecutive daily sessions. The scores form the relation:

(1, 11),\ (2, 9),\ (3, 7),\ (4, 5),\ (5, 3)

.

a

Deduce the rule connecting the session number

x

and the pain score

y

. [1]

b

Express the domain and range of this relation using inequality notation. [1]

Solutions

5QuestionDetermining Domain and Range from GraphsConcept Practice

2 marks~3 minCriterion A

A drone follows a parabolic flight path over a restricted zone. Its horizontal position

x

(in kilometres) is modelled by a continuous quadratic function, graphed with closed endpoints at

x = -3

and

x = 5

.

a

State the domain of the function in interval notation. [1]

b

The restricted zone spans

-3 \leq x \leq 5

km. Justify whether the drone's entire flight path lies within the restricted zone. [1]

Solutions

6QuestionDeriving 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

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

8QuestionSolving 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

9QuestionEvaluating 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

10QuestionChecking if Two Functions Are InversesAssessment Practice

3 marks~5 minCriterion C

The graph shows

f(x) = \sqrt{x+1}

(solid curve), the line

y = x

(dashed), and a dotted curve that is the reflection of

f

across

y = x

.

a

Explain the reflection property that connects a function to its inverse, as illustrated in this graph. [1]

b

Show that the inverse of

f(x) = \sqrt{x+1}

is

g(x) = x^2 - 1

with domain

x \geq 0

. [1]

c

The dotted curve is proposed as a model for a physical process that only produces non-negative outputs. Assess whether

g(x) = x^2 - 1

with domain

x \geq 0

is an appropriate model for this constraint. [1]

Solutions

11QuestionChecking if Two Functions Are InversesAssessment Practice

8 marks~12 minCriterion D

A pharmaceutical company stores vaccines at temperatures between

2^\circ\text{C}

and

8^\circ\text{C}

. Their international storage log records only integer Fahrenheit temperatures. The conversion formulas are

C = \dfrac{5}{9}(F - 32)

and

F = \dfrac{9}{5}C + 32

.

a

Show that the two conversion formulas are inverse functions by computing both compositions

C(F(C))

and

F(C(F))

. Clearly show each algebraic step. [3]

b

Explain how recording only integer Fahrenheit values introduces error in the corresponding Celsius temperatures. Support your explanation with a numerical example. [2]

c

The storage log shows Fahrenheit readings ranging from

35^\circ\text{F}

to

46^\circ\text{F}

. Convert these endpoints to Celsius, then advise the company whether this Fahrenheit range should be accepted as sufficient to guarantee that vaccines remain within the required

2^\circ\text{C}

to

8^\circ\text{C}

safe storage range. [3]

Solutions

12QuestionEvaluating 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

13QuestionFinding the Common RatioAssessment Practice

5 marks~8 minCriterion D

A biologist records the height of a bamboo plant over four consecutive days. The heights, in centimetres, form a geometric sequence.

Day	1	2	3	4
Height (cm)	2	6	18	54

a

State the common ratio of the sequence. [1]

b

The biologist models the height on day

n

as

H(n) = 2 \times 3^{n-1}

.

The table below shows predicted and observed heights for three days beyond the initial data.

Day	10	15	20
Predicted $H$ (cm)	39 366	9 565 938	3 486 784 401
Observed $H$ (cm)	41	67	89

Calculate the percentage error between the predicted and observed heights on day 10, giving your answer to 3 significant figures.

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%

[2]

c

Using the percentage errors evident from the table, advise the biologist whether the geometric model

H(n) = 2 \times 3^{n-1}

should be used to predict bamboo height beyond day 4. Justify your advice with one reason, in context, why the model breaks down over time. [2]

Solutions

14QuestionSolving Word Problems with Geometric SequencesAssessment Practice

8 marks~12 minCriterion C

a

State which property follows a geometric sequence and write down the common ratio. [2]

b

Write down a general rule for the value

V_n

(in USD) of the geometric property after

n

years, where

n = 1

corresponds to Year 1. Hence calculate the value of that property at the end of Year 10, giving your answer in USD to the nearest dollar. [3]

c

A colleague claims that Property A also follows a geometric sequence because its values increase each year. Calculate the ratio between each pair of consecutive terms of Property A, then justify whether the colleague's claim should be accepted or rejected, referring to what a geometric sequence requires. [3]

Solutions

15QuestionRecognizing Geometric SequencesAssessment Practice

9 marks~14 minCriterion B

The diagram shows Figures 1 to 4 of a pattern made from small equilateral triangles. Figure 1 contains 1 small triangle, Figure 2 contains 3, Figure 3 contains 6, and Figure 4 contains 10.

a

Find the number of small triangles in Figure 5 and in Figure 6. [2]

b

A student proposes the rule

T(n) = \frac{n(n+1)}{2}

for the number of small triangles in Figure

n

.

Show that

T(n) - T(n-1) = n

for all integers

n \geq 2

, and hence justify why the first differences of the sequence

1, 3, 6, 10, \ldots

are the consecutive integers

2, 3, 4, 5, \ldots

[3]

c

Using the result from part (b), prove that the ratio

\dfrac{T(n)}{T(n-1)}

is not constant for

n \geq 2

, and hence decide whether the sequence

1, 3, 6, 10, 15, \ldots

can ever be geometric. Justify your decision. [4]

Solutions

16QuestionSolving Word Problems with Geometric SequencesAssessment Practice

10 marks~15 minCriterion A

A student deposits 5000 dollars in a savings account that pays 4% compound interest per year. At the end of each year, after interest is added, the student withdraws 500 dollars. Let

B_n

denote the balance (in dollars) at the start of year

n

, so

B_1 = 5000

. The recurrence relation is

B_{n+1} = 1.04\,B_n - 500.

The diagram shows the first few values of

B_n

plotted against

n

.

a

Calculate

B_2

,

B_3

, and

B_4

, giving each answer to the nearest cent. [3]

b

Show that the substitution

C_n = B_n - 12500

transforms the recurrence relation into the geometric sequence

C_{n+1} = 1.04\,C_n

, and hence write down an explicit formula for

B_n

in terms of

n

. [4]

c

The student claims the account will never reach a zero balance. Justify whether this claim is correct, identifying the smallest value of

n

for which

B_n \leq 0

and explaining what this means for the student's savings. [3]

Solutions

17QuestionApplications in Tables and GraphsAssessment Practice

5 marks~8 minCriterion B

A city's two water-storage tanks are monitored over time.

Tank A — volume recorded at hourly intervals:

x

(hours): 1, 2, 3, 4, 5

y

(litres): 3, 7, 11, 15, 19

Tank B — volume recorded at hourly intervals:

x

(hours): 1, 2, 3, 4, 5

y

(litres): 2, 6, 12, 20, 30

a

Calculate the first differences for Tank A. Justify whether the relationship between

x

and

y

is linear or quadratic. [1]

b

Deduce an expression for

y

in terms of

x

for Tank A. [1]

c

Calculate the first and second differences for Tank B. Justify whether the relationship between

x

and

y

is linear or quadratic. [1]

d

Deduce an expression for

y

in terms of

x

for Tank B. [1]

e

Both tanks have a maximum capacity of 115 litres. Using your expressions, determine at which integer hour each tank first reaches or exceeds this capacity. Advise the city's water management team which tank requires priority monitoring, and why. [1]

Solutions

18QuestionApplications in Tables and GraphsAssessment Practice

2 marks~3 minCriterion D

Outline one real-world context where the nth term of a linear sequence is used to make predictions. The scenario involves a student saving money each week. The savings are recorded in the following rows: Week: 1, 2, 3, 4 Savings (dollars): 5, 10, 15, 20

a

Determine the nth term rule for the sequence.

b

Use the rule to predict the savings at Week 12.

c

Discuss one limitation of using this linear model for long-term savings predictions.

Solutions

19QuestionApplications in Tables and GraphsAssessment Practice

6 marks~9 minCriterion A

A scientist records the temperature of a cooling liquid every minute. The temperatures in degrees Celsius are: 90, 82, 74, 66, 58.

a

Find the nth term rule for this sequence.

b

Use your rule to predict the temperature after 10 minutes.

c

The liquid becomes unsafe at temperatures below 10°C. Use your model to determine when the temperature will first drop below 10°C.

d

Evaluate the appropriateness and impact of using this linear model for the cooling process. In your response, compare the predicted temperature at 10 minutes with the actual temperature if the cooling slows over time. Discuss at least two limitations of the model and explain the real-world impact of relying on this model for safety decisions.

Solutions

20QuestionUnderstanding 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

21QuestionDefining 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

22QuestionUnderstanding Function Notation f of xAssessment Practice

4 marks~6 minCriterion D

A mobile phone carrier charges a flat monthly fee of 15 dollars plus 0.05 dollars per minute of talk time. The monthly cost in dollars is modelled by

C(m) = 15 + 0.05m

where

m

is the number of minutes used in a month.

a

Calculate

C(200)

, the predicted monthly cost for a user who talks for 200 minutes. [1]

b

A second user budgets a maximum of 40 dollars per month. Determine the maximum number of minutes this user can talk without exceeding their budget. [1]

c

Advise the second user whether

C(m)

alone is sufficient for planning their monthly phone budget, referring to your result from part (b) and at least two limitations of the model. [2]

Solutions

23QuestionIdentifying Domain and Range from TablesAssessment Practice

4 marks~6 minCriterion D

A biologist records the number of bacteria in a culture every hour for 10 hours.

Time (hours): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Population (thousands): 0.5, 0.8, 1.3, 2.1, 3.4, 5.5, 8.9, 14.4, 23.3, 37.7, 61.0

a

State the domain of this model. [1]

b

State the range of this model. [1]

c

The biologist uses this model to predict the bacterial population at

t = 20

hours. Advise the biologist whether this prediction should be trusted, referring to the domain identified in part (a) and to the biological conditions of the culture. [2]

Solutions

24QuestionIdentifying 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

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