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Number Patterns and Arithmetic Sequences — 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:
These points are plotted on a graph with on the horizontal axis and on the vertical axis.
| Month () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Users (, hundreds) | 4 | 7 | 10 | 13 |
These points are plotted on a graph with on the horizontal axis and on the vertical axis.
a
Explain how the graph shows that the common difference is . [1]
b
Deduce the formula for , the number of users (in hundreds) in month . [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]
Solutions
2QuestionFinding 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 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]
Solutions
3QuestionSolving Problems Using Arithmetic SequencesAssessment Practice
16 marks~24 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 represent the total number of blocks needed to build a staircase with steps.
Question
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 represent the total number of blocks needed to build a staircase with steps.
Question
a
Calculate , , , and . Show clearly how each total is formed from the blocks in each step. [4 marks]
b
Predict a general formula for in terms of . Your answer should:
- Identify the pattern in the totals,
- Express the relationship using algebra,
- Clearly define what and represent. [4 marks]
- Identify the pattern in the totals,
- Express the relationship using algebra,
- Clearly define what and represent. [4 marks]
c
Justify your formula by showing that can be written as the sum of the first terms of an arithmetic series. In your response:
- Identify the first term ,
- Identify the common difference ,
- Write the th term of the sequence,
- Use the arithmetic series sum formula,
- Simplify your expression to match your formula from part (b). [6 marks]
- Identify the first term ,
- Identify the common difference ,
- Write the 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: . 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 th step,
- Total blocks in steps. [2 marks]
(e) Explain why clear notation is important when solving this problem, especially the difference between:
- Blocks in the th step,
- Total blocks in steps. [2 marks]
Solutions
4QuestionDeriving 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 th term of this arithmetic sequence is .
Week 1: 15 — Week 2: 22 — Week 3: 29 — Week 4: 36
A friend claims the th term of this arithmetic sequence is .
a
Calculate the values predicted by for Weeks 1 to 4. [2]
b
Deduce, with supporting calculations, whether the formula correctly models the student's savings. [2]
c
The student has a savings goal of 100 dollars in a single week. Using the correct th term formula, justify whether this goal will ever be reached exactly under the same saving pattern. [2]
Solutions
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