Number Patterns and Arithmetic Sequences
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Number Patterns and Arithmetic Sequences — Free MYP4 Mathematics (Extended) Practice 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 an=a1+(n−1)da_n = a_1 + (n - 1)d, where ana_n is the nth term, a1a_1 is the first term, nn is the term number, and dd 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 Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n), where SnS_n is the sum of the first n terms, a1a_1 is the first term, and ana_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.
Question diagram

Solutions

2QuestionFinding 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.

Week12345
Balance (dollars)1015202530
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]
Question diagram

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 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]

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 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]

Solutions

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