Sequences Patterns and Functions — Free IB MYP5 Mathematics (Standard) Questions

1QuestionDeriving the nth Term of an Arithmetic SequenceConcept Practice

2 marks~3 minCriterion C

A theatre adds more seats to each row from the front to the back. The number of seats in successive rows forms the arithmetic sequence: 7, 11, 15, 19, …

a

Deduce the common difference of the sequence. [1]

b

A theatre manager claims that row 30 will have exactly 119 seats. Derive the formula for the

n

th term of the sequence, then justify whether the manager's claim is correct. [1]

Solutions

2QuestionChecking if Two Functions Are InversesConcept Practice

2 marks~3 minCriterion C

A traveller flying from London to New York checks a weather app showing the forecast in Celsius. On arrival, all temperatures are displayed in Fahrenheit.

The conversion formulas are

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

and

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

.

a

Show that applying both formulas in succession to

C = 20

returns the original value. [1]

b

Advise the traveller whether these two formulas can be used reliably for temperature conversion in both directions. Justify your answer using the result from part (a). [1]

Solutions

3QuestionWriting Inequality Notation for Domain and RangeConcept Practice

2 marks~3 minCriterion A

A drone delivery service operates along a straight route. Its altitude function

f(x)

is graphed below, where

x

represents horizontal distance (km) from the depot.

The graph shows an open circle at

x = -3

and a closed circle at

x = 5

.

Write the domain of

f

using inequality notation. [2]

Solutions

4QuestionSolving 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

5QuestionSolving Problems Using Arithmetic SequencesAssessment Practice

16 marks~24 minCriterion D

A warehouse uses a system of stacked storage boxes. The heights of the boxes form an arithmetic sequence: Box 1 has a height of 50 cm, and each box above it is 3 cm shorter than the one below.

The sequence of box heights (cm) begins: 50, 47, 44, 41, …

a

Calculate the heights of Box 4, Box 5, and Box 6. [2]

b

State the first term and common difference of the sequence, then deduce a formula for

h_n

, the height (in cm) of the

n

th box. [3]

c

Show that the 15th box has a height of 8 cm, and calculate the height of the 10th box. [3]

d

The total stacked height must not exceed 350 cm. Using the arithmetic series sum formula, determine the maximum number of boxes that can be stacked. Justify your answer. [4]

e

The warehouse manager plans to extend the sequence to 20 boxes. Advise the manager whether this plan can be implemented, supporting your advice with both a mathematical calculation and a real-world explanation. [4]

Solutions

6QuestionRecognizing and Continuing Arithmetic SequencesAssessment Practice

2 marks~3 minCriterion A

A multi-storey car park numbers its spaces using an arithmetic sequence. The first three spaces on a new floor are numbered 102, 108, and 114.

The floor has capacity for exactly 20 spaces, and the car park manager claims that space number 222 will appear on this floor.

a

Deduce the number of the 4th parking space. [1]

b

Advise the car park manager whether space 222 will appear on this floor, supporting your advice with mathematical reasoning. [1]

Solutions

7QuestionEvaluating 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

8QuestionChecking if Two Functions Are InversesAssessment Practice

4 marks~6 minCriterion D

A traveller converts euros to US dollars using

f(x) = 1.08x

, then back to euros using

g(x) = \dfrac{x}{1.08}

, where

x

represents the amount in the respective currency.

a

Prove that

f

and

g

are inverse functions by computing

f(g(x))

and

g(f(x))

. Show all steps. [2]

b

A bank applies a 2% fee on every conversion and rounds each result to the nearest cent. Advise the traveller whether

f

and

g

can be relied upon as exact inverses under these real banking conditions, justifying your answer with a specific numerical example. [2]

Solutions

9QuestionEvaluating Composite FunctionsAssessment Practice

5 marks~8 minCriterion A

A company models its profit

P(t)

(in thousands of dollars) using the composite function

P(t) = R(C(t))

, where

t

is time in years. The production cost function is

C(t) = 2t + 5

and the revenue function is

R(x) = 3x - 4

, where

x

represents production cost (both in thousands of dollars).

a

Evaluate

P(5)

. [2]

b

The company considers a profit of at least 40 thousand dollars at

t = 5

to be financially viable. Justify whether this target is met. [1]

c

Discuss one strength and one limitation of using this composite function to forecast business profit. [2]

Solutions

10QuestionApplications 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

11QuestionCreating a Rule for the nth TermAssessment Practice

6 marks~9 minCriterion C

A landscape architect designs a series of square garden terraces. The total paved area (in m²) after each terrace is added follows a quadratic sequence:

$n$	1	2	3	4	5
$u_n$	2	6	12	20	30

A student proposes the rule

u_n = n^2 + n

for the

n

th term.

a

Verify that the rule

u_n = n^2 + n

produces the correct values for

n = 1, 2, 3, 4, 5

. [2]

b

Deduce the total paved area after the 10th terrace is added. [1]

c

The architect has a maximum budget allowing 182 m² of paving. Advise the architect whether a 13th terrace can be added within this budget, justifying your answer using the proposed rule. [3]

Solutions

12QuestionApplications 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

13QuestionApplications in Tables and GraphsAssessment Practice

4 marks~6 minCriterion D

A student records the battery percentage of a smartphone during continuous video playback.

Time (hours)	0	2	4	6	8
Battery (%)	100	75	50	25	0

The student models battery drain using:

B(t) = 100 - 12.5t

where

B

is the battery percentage and

t

is the time in hours.

a

Calculate the predicted battery percentage after 3 hours. [1]

b

The student claims the model can be used to schedule a 90-minute video call starting at

t = 6

hours. Using the model, determine whether the battery will last the full call, then advise the student whether the linear model is reliable for this scheduling decision. [3]

Solutions

14QuestionUnderstanding 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

15QuestionDefining 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

16QuestionDefining Piecewise and Step Functions IntroductoryAssessment Practice

2 marks~3 minCriterion A

A delivery drone travels along a straight route. Its speed

v

(in m/s) over time

t

(in seconds) is modelled by the graph shown.

The graph shows:
- a horizontal segment from

t = 0

to

t = 4

at

v = 3

- a segment with slope

2

from

(4,\ 3)

to

(8,\ 11)

Construct the piecewise function

v(t)

using correct mathematical notation. [2]

Solutions

17QuestionUnderstanding Function Notation f of xAssessment Practice

6 marks~9 minCriterion D

A civil engineer models the cost, in thousands of dollars, of a concrete beam of length

x

metres using

f(x) = 1.2x + 50

. A 15% safety surcharge gives the total cost function

g(f(x)) = 1.15\, f(x)

.

The measured length of a beam is 12 metres, with a possible error of

\pm 0.5

metres.

a

Calculate

g(f(11.5))

and

g(f(12.5))

to determine the range of total costs for this beam. [2]

b

Deduce whether the linear model

f(x)

remains appropriate when

x = 100

metres, referring to how material and structural behaviour affect costs at large beam lengths. [2]

c

The company must submit a single fixed-cost bid for this beam. Advise the engineer which value to bid, justifying your recommendation using the range found in part (a) and the financial consequence of the alternative choice. [2]

Solutions

18QuestionIdentifying Discrete and Continuous FunctionsAssessment Practice

4 marks~6 minCriterion B

A mobile data analyst records the number of text messages sent per hour during a network stress test:

h

(hour):

1, \ 2, \ 3, \ 4

m

(messages):

5, \ 8, \ 13, \ 20

a

Calculate the next three terms of the sequence. [1]

b

Deduce a formula for

m

in terms of

h

that fits all given data points. [2]

c

The analyst proposes using this model to predict message counts at non-integer hours such as

h = 1.5

. Advise the analyst whether this proposal is appropriate, justifying your answer with reference to the domain of the function. [1]

Solutions

19QuestionIdentifying Domain and Range from TablesAssessment Practice

6 marks~9 minCriterion D

A city's water treatment plant records incoming flow rate (in

\text{m}^3/\text{h}

) every hour for 24 hours.

Hour: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

Flow (

\text{m}^3/\text{h}

): 120, 115, 130, 145, 160, 180, 200, 210, ?, ?, ?, 195, 170, 155, 140, 130, 125, 135, 150, 170, 190, 185, 160, 140

a

State the domain of the function represented by the valid readings only. Write your answer in set notation. [2]

b

Deduce the range of the flow rates using the valid readings only. Write your answer in set notation. [2]

c

Advise the plant operators whether the incomplete data are sufficient to plan for peak demand. In your response:
— identify which hours are missing and explain why they are significant for peak demand;
— state one assumption needed to estimate the missing values and evaluate how that assumption affects the reliability of capacity planning. [2]

Solutions

20QuestionWriting Inequality Notation for Domain and RangeAssessment Practice

4 marks~6 minCriterion C

A piecewise function

f(x)

is defined by:

f(x) = \begin{cases} x^2 - 4, & x < 1 \\ -\dfrac{1}{3}x - \dfrac{2}{3}, & 1 \leq x \leq 4 \end{cases}

The parabolic piece has an open circle at

(1,\ -3)

; the linear piece has closed circles at

(1,\ -1)

and

(4,\ -2)

.

a

Write the domain of

f(x)

in inequality notation. [1]

b

Justify why

x = 1

is included in the domain of

f(x)

, despite the open circle on the parabolic piece. [1]

c

A student claims: "Because $f(x)$ is discontinuous at $x = 1$ , the point $x = 1$ must be excluded from the domain." Critique this claim, using the left-hand limit and the value of

f(1)

to support your reasoning. [2]

Solutions

Sequences Patterns and Functions — Free MYP5 Mathematics (Standard) Practice Questions