Arithmetic pricing model

In this problem, we explore how an arithmetic sequence can model a simple pricing structure. This connects directly to the HSC Sequences topic, where recognising linear growth patterns unlocks quick formula-building and inequality solving. By working through this example, you’ll practise translating a word problem into a sequence formula and then using it to answer both evaluation and budget-limit questions.

Problem Statement

An installer charges $400 for the first window and $100 for each additional window.

• Write the total cost for 1, 2, 3, and 4 windows.
• Find a formula for the total cost $A_n$ for $n$ windows.
• Find $A_{15}$.
• If the budget is strictly less than $10,000, what is the maximum number of windows?

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Hints

Model $A_n$ as an arithmetic sequence with first term $400$ and difference $100$.

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Solutions

We begin by listing the costs for the first few numbers of windows. The first window costs $400, and every extra window adds $100:

$$C_1 = 400,\qquad C_2 = 400+100 = 500,\qquad C_3 = 600,\qquad C_4 = 700.$$

The cost of $n$ windows follows an arithmetic progression with first term $a = 400$ and common difference $d = 100$. The $n$th term of an arithmetic sequence is $A_n = a + (n-1)d$, so

$$A_n=400+(n-1)100=100n+300.$$

This compact formula now lets us answer any number of windows. Substituting $n = 15$ gives the total cost directly: So $A_{15}=1800$.

Finally, we translate the budget restriction into an inequality. The total cost must be strictly less than $10,000, so

$$100n+300<10000 \implies n<97,$$

hence the maximum integer number of windows is $n=96$.

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Takeaways

• Recognise that a constant addition ($100 per window) signals an arithmetic sequence; the first term and common difference are all you need.
• Build the explicit formula $A_n = a + (n-1)d$ and then use it for both simple evaluations and inequality constraints.
• When an inequality gives a non-integer bound (like $n < 97$), always round down to the nearest integer to stay within the strict budget limit.

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Further Readings

HSC Sequences, HSC Vectors, HSC Differential Equations, HSC Probability