Problem 78 - A Graphical Approach to College Algebra

Question

Solve each problem. (Modeling) Spending on Food The average family in the United States spends \(\$ 150\) 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

The family spends $150 in the fourth week.

Step 1: Identifying the Sequence Type

The exercise presents a scenario where spending on food is constant every week: $150 per week. This defines an arithmetic sequence where each term is the same, with a common difference of 0. Thus, it is more fitting to consider this sequence as a constant sequence.

Step 2: Define the General Term

Since the spending is constant each week, the sequence can be defined as \(a_n = 150\). This means that for any week number \(n\), the spending amount is $150.

Step 3: Calculate the Specific Term

To find the spending amount for the fourth week, substitute \(n = 4\) into the general term: \(a_4 = 150\).

Step 4: Interpret the Result

The result \(a_4 = 150\) indicates that the family spends $150 on food during the fourth week, consistent with their weekly spending pattern.

Key Concepts

Constant Sequence

A constant sequence is a simple type of sequence where all terms are the same. This means that the value of each term in the sequence does not change as you progress through the terms. For example, if a family spends \(150\) dollars on food every week, this is represented as a constant sequence.The defining characteristic of a constant sequence is that its common difference is zero. A common difference in a sequence is the amount you add to get from one term to the next. Since the spending each week does not change, you're adding zero each time to reach the next term. Think of the sequence \(150, 150, 150, ...\), where each term remains constant. This is why the constant sequence is incredibly predictable and easy to model. It simplifies the process of finding any term in the sequence, as you'll see in the next section.

General Term

The general term of a sequence provides a formula to find any term without having to list all the previous ones. This is particularly handy when dealing with sequences of large numbers or many elements. In our case, the general term is simply a mathematical way to express the constant spending: \(a_n = 150\). The formula shows that no matter what the week number \(n\) is, the amount spent remains the same - \(150\) dollars.This formula allows you to calculate the spending for any future week by plugging in the week number for \(n\). For instance, to find the spending in the tenth week, you apply the formula: \(a_{10} = 150\). You will arrive at the same number, confirming the spending does not vary week to week.

Arithmetic Progression

An arithmetic progression is a sequence where each term after the first is obtained by adding a fixed, non-zero number, known as the "common difference," to the previous term. Typically, this involves sequences where each term grows or decreases by a constant amount.However, in a constant sequence like ours, the special case is that the common difference is zero. While this might seem contrary to what makes an arithmetic progression, the term 'arithmetic sequence' still applies because no term deviates from the starting value.When understanding arithmetic sequences, the general formula applies: \(a_{n} = a_{1} + (n-1)\cdot d\), where \(a_{1}\) is the first term and \(d\) is the common difference. For our sequence:

• The first term \(a_{1} = 150\),
• The common difference \(d = 0\),

This reaffirms that the formula leads to each term being equal to the initial one, demonstrating consistency across weeks with no increase or decrease in spending.