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Sequences Patterns and Functions

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

1QuestionDeriving the nth Term of an Arithmetic SequenceConcept Practice
3 marks~5 minCriterion A
A city's public bike-share programme tracks registered users over four consecutive months. The data form an arithmetic sequence:

Month (nn)1234
Users (unu_n, hundreds)471013


These points are plotted on a graph with nn on the horizontal axis and unu_n on the vertical axis.
a
Explain how the graph shows that the common difference is d=3d = 3. [1]
b
Deduce the formula for unu_n, the number of users (in hundreds) in month nn. [1]
c
The programme's funding is renewed only if registered users exceed 5 000 before the end of month 18. Justify whether this funding condition is met. [1]
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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 nn of the term in this arithmetic sequence that has value ana_n, expressing nn in terms of ana_n, a1a_1, and dd. Show all algebraic steps clearly. [2]

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3QuestionWriting 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)(1, 11),\ (2, 9),\ (3, 7),\ (4, 5),\ (5, 3).
a
Deduce the rule connecting the session number xx and the pain score yy. [1]
b
Express the domain and range of this relation using inequality notation. [1]

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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 SnS_n represent the total number of blocks needed to build a staircase with nn steps.

Question
a
Calculate S1S_1, S2S_2, S3S_3, and S4S_4. Show clearly how each total is formed from the blocks in each step. [4 marks]
b
Predict a general formula for SnS_n in terms of nn. Your answer should:

- Identify the pattern in the totals,
- Express the relationship using algebra,
- Clearly define what nn and SnS_n represent. [4 marks]
c
Justify your formula by showing that SnS_n can be written as the sum of the first nn terms of an arithmetic series. In your response:

- Identify the first term aa,
- Identify the common difference dd,
- Write the nnth 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: Sn=n2[2a+(n−1)d]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 nnth step,
- Total blocks in nn steps. [2 marks]

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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 nnth term of this arithmetic sequence is un=8n+7u_n = 8n + 7.
a
Calculate the values predicted by un=8n+7u_n = 8n + 7 for Weeks 1 to 4. [2]
b
Deduce, with supporting calculations, whether the formula un=8n+7u_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 nnth term formula, justify whether this goal will ever be reached exactly under the same saving pattern. [2]

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6QuestionFinding the Common DifferenceAssessment Practice
6 marks~9 minCriterion D
A bank offers a loan of 10 000 dollars to be repaid over 10 months. The monthly repayments follow an arithmetic sequence. The first payment is 1 200 dollars and each subsequent payment decreases by a fixed amount, so that the tenth payment is 300 dollars.
a
Show that the common difference of the repayment sequence is −100-100 dollars. [2]
b
Calculate the total amount repaid over the 10 months. [1]
c
Advise the bank whether this repayment plan should be offered to borrowers, discussing what the total repayment implies about interest or fees and how real-world factors such as variable interest rates or late-payment penalties affect the reliability of this model. [3]
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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 ff doubles the strength, then filter gg adds 1 unit. Formally, f(x)=2xf(x) = 2x and g(x)=x+1g(x) = x + 1, giving composite function h(x)=g(f(x))h(x) = g(f(x)). The simulation begins at strength x0=1x_0 = 1 and updates by xn+1=h(xn)x_{n+1} = h(x_n).
a
Calculate x1x_1, x2x_2, x3x_3, and x4x_4. [2]
b
Deduce a general formula for xnx_n in terms of nn. [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]

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8QuestionEvaluating Composite FunctionsAssessment Practice
2 marks~3 minCriterion A
A surveyor uses a 3-4-5 right triangle to set a reference angle θ\theta, where θ\theta is the angle opposite the side of length 3.

The functions f(x)=sin⁡(x°)f(x) = \sin(x°) and g(x)=x+30g(x) = x + 30 model how the surveyor adjusts bearing angles.

Calculate the value of f(g(θ))f(g(\theta)), giving your answer to 3 significant figures. [2]
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9QuestionChecking if Two Functions Are InversesAssessment Practice
3 marks~5 minCriterion C
The graph shows h(x)=12x+2h(x) = \frac{1}{2}x + 2 (solid line), the line y=xy = x (dashed), and three points claimed to belong to a function p(x)p(x): (0, −4)(0,\,-4), (2, 0)(2,\,0), and (4, 4)(4,\,4).
a
Calculate the coordinates of the points on h(x)h(x) at x=0x = 0, x=2x = 2, and x=4x = 4. [1]
b
Deduce the coordinates that the inverse of h(x)h(x) must pass through by reflecting your points from (a) across the line y=xy = x. [1]
c
Justify whether p(x)p(x) is the inverse of h(x)h(x), referring to your results from (b). [1]
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10QuestionConcept of Inverse FunctionsAssessment Practice
6 marks~9 minCriterion D
A car's fuel efficiency is modelled by the function

f(x)=x12f(x) = \frac{x}{12}

where xx is the distance travelled in kilometres (km) and f(x)f(x) is the fuel used in litres (L). The model assumes constant driving conditions throughout any journey.
a
Determine the inverse function f−1(x)f^{-1}(x) and explain what it represents in this context. [2]
b
A driver has 30 L of fuel available. Using f−1(x)f^{-1}(x), calculate the maximum distance the driver can travel. [2]
c
The driver plans a 400 km motorway journey followed by a 20 km urban leg. Advise the driver whether 35 L of fuel is sufficient for the full journey, justifying your answer with reference to the model's assumption of constant driving conditions. [2]
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11QuestionSolving Word Problems with Geometric SequencesAssessment Practice
8 marks~12 minCriterion A
A student invests 500 dollars in a savings account that pays 6% interest compounded annually. The amount in the account after nn complete years is modelled by
A(n)=500×(1.06)n.A(n) = 500 \times (1.06)^n.
a
Show that the amount in the account after 7 complete years is greater than 750 dollars. [2]
b
Find the minimum number of complete years it takes for the amount in the account to first exceed 1000 dollars. You must show a method using logarithms. [3]
c
A second account also starts with 500 dollars and offers 4% interest compounded quarterly, so that the amount after nn complete years is
B(n)=500×(1.01)4n.B(n) = 500 \times (1.01)^{4n}.
After 10 complete years, account AA earns interest at its annual rate and account BB earns interest at its quarterly rate throughout. Advise the student which account to choose, justifying your recommendation using the amounts in each account after 10 complete years. Give monetary values in dollars to 2 decimal places. [3]
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12QuestionFinding the Common RatioAssessment Practice
6 marks~9 minCriterion B
The diagram shows Figures 1 to 4 of a pattern made from small equilateral triangles. Figure 1 has 1 triangle, Figure 2 has 4 triangles, Figure 3 has 9 triangles, and Figure 4 has 16 triangles.
a
Write down the number of small triangles in Figure 5. [1]
b
Find a rule for the number of small triangles TT in Figure nn. Hence find the value of nn for which Figure nn contains exactly 144 small triangles. [3]
c
A student claims that no figure in this pattern can contain exactly 50 small triangles. Justify whether the student is correct. [2]
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13QuestionSolving Word Problems with Geometric SequencesAssessment Practice
6 marks~9 minCriterion D
A bacteria culture initially contains 500 bacteria. The population doubles every 3 hours. The table below shows the model's predicted population and the recorded actual population at three checkpoints.

Time (hours)369
Predicted (bacteria)100020004000
Actual (bacteria)95018003500
a
Write down the general term unu_n of the geometric sequence that models the population at the end of each 3-hour period, where n=1n = 1 corresponds to the end of hour 3. [1]
b
Find the population predicted by the model after 15 hours. Give your answer in exact form. [2]
c
The laboratory records show that the actual population after 15 hours is 11 200 bacteria. Using the table and your answer to part (b), assess whether the geometric model is suitable for long-term population predictions, justifying your answer with reference to the data. [3]
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14QuestionSolving Word Problems with Geometric SequencesAssessment Practice
6 marks~9 minCriterion C
A bacterial culture grows so that its population triples every hour. At t=0t = 0 hours the population is 200200 bacteria. The population at time tt hours is modelled by
B(t)=200×3t.B(t) = 200 \times 3^{t}.
a
Write down the value of B(0)B(0) and hence state what the number 200200 represents in the model. [1]
b
A second culture starts at t=0t = 0 with a population of 5050 bacteria and quadruples every hour, so its population is modelled by
C(t)=50×4t.C(t) = 50 \times 4^{t}.
Find the value of tt at which the two cultures have equal populations. Give your answer to 3 significant figures. [3]
c
A scientist claims that culture BB will always have a larger population than culture CC for every hour after they are equal. Advise the scientist whether this claim should be accepted, using the growth rates of the two models to support your answer. [2]
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15QuestionApplications 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:

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

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

Tank B — volume recorded at hourly intervals:

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

yy (litres): 2, 6, 12, 20, 30
a
Calculate the first differences for Tank A. Justify whether the relationship between xx and yy is linear or quadratic. [1]
b
Deduce an expression for yy in terms of xx for Tank A. [1]
c
Calculate the first and second differences for Tank B. Justify whether the relationship between xx and yy is linear or quadratic. [1]
d
Deduce an expression for yy in terms of xx 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]

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16QuestionApplications 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)02468
Battery (%)1007550250


The student models battery drain using:
B(t)=100−12.5tB(t) = 100 - 12.5t
where BB is the battery percentage and tt 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=6t = 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]
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17QuestionFinding Term Position Given ValueAssessment Practice
7 marks~11 minCriterion A
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]
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18QuestionDefining 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]
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19QuestionUnderstanding 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 a1=2a_1 = 2 and the recurrence relation an+1=f(an)a_{n+1} = f(a_n), where f(x)=3x−1f(x) = 3x - 1.

The first four hourly populations are:
a1=2a_1 = 2, a2=5a_2 = 5, a3=14a_3 = 14, a4=41a_4 = 41
a
Calculate f(f(f(2)))f(f(f(2))). [2]
b
Deduce a general formula for ana_n in terms of nn. [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]

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20QuestionUnderstanding 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 xx metres using f(x)=1.2x+50f(x) = 1.2x + 50. A 15% safety surcharge gives the total cost function g(f(x))=1.15 f(x)g(f(x)) = 1.15\, f(x).

The measured length of a beam is 12 metres, with a possible error of ±0.5\pm 0.5 metres.
a
Calculate g(f(11.5))g(f(11.5)) and g(f(12.5))g(f(12.5)) to determine the range of total costs for this beam. [2]
b
Deduce whether the linear model f(x)f(x) remains appropriate when x=100x = 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]
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21QuestionDefining Piecewise and Step Functions IntroductoryAssessment Practice
2 marks~3 minCriterion A
A delivery drone travels along a straight route. Its speed vv (in m/s) over time tt (in seconds) is modelled by the graph shown.

The graph shows:
- a horizontal segment from t=0t = 0 to t=4t = 4 at v=3v = 3
- a segment with slope 22 from (4, 3)(4,\ 3) to (8, 11)(8,\ 11)

Construct the piecewise function v(t)v(t) using correct mathematical notation. [2]
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22QuestionApplications in Real-World GraphsAssessment Practice
2 marks~3 minCriterion D
A ball is thrown vertically upward from a rooftop. Its height above the ground, hh metres, is modelled by h(t)=−5t2+20t+15h(t) = -5t^2 + 20t + 15, where tt is the time in seconds after release. A safety regulation states that any object above 30 metres poses a risk to overhead cables.

Justify whether the ball breaches the safety regulation of 30 metres, using the function to support your conclusion. [2]
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23QuestionIdentifying Domain and Range from TablesAssessment Practice
6 marks~9 minCriterion A
A city's water treatment plant records incoming flow rate (in m3/h\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 (m3/h\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]
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24QuestionWriting Inequality Notation for Domain and RangeAssessment Practice
4 marks~6 minCriterion C
A piecewise function f(x)f(x) is defined by:

f(x)={x2−4,x<1−13x−23,1≤x≤4f(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)(1,\ -3); the linear piece has closed circles at (1, −1)(1,\ -1) and (4, −2)(4,\ -2).
a
Write the domain of f(x)f(x) in inequality notation. [1]
b
Justify why x=1x = 1 is included in the domain of f(x)f(x), despite the open circle on the parabolic piece. [1]
c
A student claims: "Because f(x)f(x) is discontinuous at x=1x = 1, the point x=1x = 1 must be excluded from the domain." Critique this claim, using the left-hand limit and the value of f(1)f(1) to support your reasoning. [2]
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