Problem 86 - A Graphical Approach to Precalculus with Limits

Question

The average family in the United States spends 150 dollars on food per week. Write a general term \(a_{n}\) for a sequence that gives the spending on food after \(n\) weeks. Find \(a_{4}\) and interpret the result.

Step-by-Step Solution

Answer

After 4 weeks, the family spends 600 dollars on food.

Step 1: Understanding the Problem

We need to construct a sequence that represents the weekly spending on food by the average family over multiple weeks. We are given that the weekly expenditure is 150 dollars.

Step 2: Define the General Term

The sequence represents cumulative spending over multiple weeks. Since each week's spending is constant, the general term of the sequence, which represents the total amount spent after \(n\) weeks, is \(a_n = 150n\).

Step 3: Calculate \(a_4\)

To find \(a_4\), plug in \(n = 4\) into the general term: \(a_4 = 150 \times 4 = 600\).

Step 4: Interpret the Result

The result \(a_4 = 600\) represents the total amount spent on food by the average family after 4 weeks, which is 600 dollars.

Key Concepts

Weekly Spending

Weekly spending refers to the consistent amount of money allocated or spent by an individual or family in a typical week. In this case, the exercise specifies that an average U.S. family spends $150 on food per week. Understanding weekly spending is crucial because it forms the foundation of budgeting and financial planning for many households.

• Weekly spending helps in assessing cash flow on a regular basis.
• It enables families to identify areas that may require adjustments or savings.

This constant expenditure creates a predictable pattern, which is vital for solving problems related to arithmetic sequences, as seen in this exercise. With a fixed weekly expenditure, the arithmetic sequence becomes clear, allowing us to use mathematical tools to analyze cumulative costs over more extended periods.

Cumulative Expenditure

Cumulative expenditure refers to the total amount of money spent over a period of time. It's like a running tally of all the spending done cumulatively over successive weeks. In the context of this exercise, if a family spends $150 each week, their cumulative expenditure after a given number of weeks is calculated by adding up all the weekly expenses.
Consider the weekly spending as a sequence where each term represents spending for one week:

• Week 1: $150
• Week 2: $150 + $150 = $300
• Week 3: $300 + $150 = $450
• Week 4: $450 + $150 = $600

This progression shows how cumulative expenditure grows linearly. This pattern is a direct result of weekly spending being consistent, making it a useful scenario to practice with sequences, particularly arithmetic ones.

General Term of a Sequence

The general term of a sequence is a formula that allows us to find the value of any term within that sequence without having to list all preceding terms. In arithmetic sequences, like daily or weekly spending scenarios, this is especially handy. Given a regular pattern of increasing or decreasing values, deriving the general term provides a quick and efficient method to calculate a desired outcome.
For the average U.S. family's weekly spending, every week adds \(150 to the total spent. This information allows us to write a general term for the sequence:

• Let the general term be represented as \(a_n\).
• Since each week introduces a constant of \)150, the pattern follows an arithmetic rule.
• The formula becomes \(a_n = 150n\), where \(n\) signifies the number of weeks.

This formula essentially simplifies the calculation of cumulative expenditure after any given week by multiplying the constant weekly spending by the number of weeks, allowing students to easily find things like \(a_4\), which represents the spending after four weeks.