Problem 45 - Algebra for College Students

Question

To assess the economic impact of a factory on a community, economists consider the annual amount the factory spends in the community, then the portion of the money that is respent in the community, then the portion of the respent money that is respent in the community, and so on. Suppose a garment manufacturer spends \(\$ 1\) million annually in its community and \(80 \%\) of all money received in the community is respent in the community. Find the first four terms of the economic impact sequence.

Step-by-Step Solution

Answer

The first four terms are 1 million, 0.80 million, 0.64 million, and 0.512 million.

Step 1: Identify the Initial Spend

The factory initially spends \( 1 \text{ million} \text{ dollars} \) annually in the community. This establishes the first term of the sequence.

Step 2: Determine the Respent Percentage

Given that \( 80\text{%} \text{ of the money is respent in the community, we can convert this to a decimal fraction: \) \ 0.80 \.

Step 3: Calculate the First Term

The first term of the sequence is the initial amount: \( 1 \text{ million} \)

Step 4: Calculate the Second Term

Multiply the first term by \( 0.80 \) to find the second term: \( 1 \text{ million} \times 0.80 = 0.80 \text{ million} \)

Step 5: Calculate the Third Term

Multiply the second term by \( 0.80 \): \( 0.80 \text{ million} \times 0.80 = 0.64 \text{ million} \)

Step 6: Calculate the Fourth Term

Multiply the third term by \( 0.80 \): \( 0.64 \text{ million} \times 0.80 = 0.512 \text{ million} \)

Key Concepts

Initial Spend

The initial spend is the starting point of our sequence in understanding economic impact.
In this example, the garment manufacturer spends \(1 \text{ million} \text{ dollars}\) annually in the community.
This initial spend is considered the first term of our geometric series.
Understanding the amount spent initially is crucial because it sets the foundation for all subsequent calculations.
Every time the money cycles through the community, we start from this base amount.

Respent Percentage

The respent percentage refers to the portion of money that is spent again within the community.
In this example, \(80\text{\text{%}}\) of the spent money is respent. This percentage is crucial for determining the subsequent terms in our sequence.
The respent percentage is converted to a decimal format for ease of calculation: \(0.80\).
This decimal form is then used to multiply the initial spend and each following term to find the next term in the sequence.

Terms of a Sequence

In a geometric sequence, each term is derived from the previous one by multiplying with a constant.
Here, we started with the initial spend of \(1 \text{ million} \text{ dollars}\) and used the respent percentage of \(0.80\).

• The first term is \(1 \text{ million}\)
• The second term is \(1 \text{ million} \times 0.80 = 0.80 \text{ million}\)
• The third term is \(0.80 \text{ million} \times 0.80 = 0.64 \text{ million}\)
• The fourth term is \(0.64 \text{ million} \times 0.80 = 0.512 \text{ million}\)

By following these steps, we can continue the sequence to find further terms if needed.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant.
In the context of economics, it models how money circulates within a community.
For our example:

• The series starts with \(1 \text{ million}\)
• Each following term is \(0.80 \text{ times the previous term}\)

The resulting series describes how the initial \(1 \text{ million} \text{ dollars}\) spent by the factory generates further economic activity.
This concept is vital in assessing the economic impact because it considers not just the initial spend, but the multiplication effect of respent money throughout the community.