Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

A hardware store makes a profit of \(\$ 30,000\) during its first year. The store owner sets a goal of increasing profits by 5000 dollar each year for 4 years. Assuming that this goal is met, find the total profit during the first 5 years of business.

Step-by-Step Solution

Answer

The total profit during the first 5 years of business is $200,000.

Step 1: Identify the Arithmetic Sequence

In this case, the arithmetic sequence is the profit made each year, which starts at $30,000 and increases by $5,000 each year. So the arithmetic sequence is $30,000, $35,000, $40,000, $45,000, and $50,000.

Step 2: Calculate the Sum of the Arithmetic Sequence

Using the formula for the sum of the first n terms of an arithmetic sequence, which is \(S_n = n/2 * (a + l)\), where \(n\) is the number of terms (5), \(a\) is the first term (30,000), and \(l\) is the last term (50,000). By plugging in these values, we get \(S_5 = 5/2 * (30,000 + 50,000)\).

Step 3: Compute the Total Profit

The result from Step 2 gives us the total profit over the five years. So, multiply and add the terms as prescribed in the formula: \(S_5 = 5/2 * (80,000) = 5 * 40,000 = $200,000\).

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. In our example, the sequence begins with a first-year profit of \(30,000. Each year, the profit increases by \)5,000. Thus, the sequence for the profits over five years is: \(30,000, \)35,000, \(40,000, \)45,000, and \(50,000.

To find the sum of this sequence, we utilize the formula for the sum of an arithmetic sequence:

• \(S_n = \frac{n}{2} \times (a + l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.

Plugging in the values:\(S_5 = \frac{5}{2} \times (30,000 + 50,000)\).

First, calculate the sum within the parentheses: \(30,000 + 50,000 = 80,000\). Then, divide the number of terms, 5, by 2, to get 2.5. Now multiply these together: \(2.5 \times 80,000 = 200,000\).

Therefore, the total profits for five years, following this sequence, are \)200,000.

Profit Calculation

Profit calculation, in this context, involves understanding how anticipated increases in profits contribute to total earnings over a set time period. The store intends to increase its profits by $5,000 each year for four years, starting from an initial profit of $30,000.

Consider each stepped increase as an additional term in an arithmetic sequence. Accordingly, you begin with $30,000 as the starting point. Then, for subsequent years, add the projected increase ($5,000) each year to derive the profit for that year.

• Year 1: $30,000
• Year 2: $35,000
• Year 3: $40,000
• Year 4: $45,000
• Year 5: $50,000

This cumulative process showcases how a gradual increase in profit affects overall financial success. It's essential for business planning and helps set realistic financial goals.

Business Mathematics

Business mathematics is a critical skill in managing and anticipating financial performance. It entails the use of mathematical methods to solve practical business issues, such as predicting profits.

By applying the arithmetic sequence formula, as in our hardware store example, business owners can project future earnings and make informed decisions.

• It helps in aligning financial goals with business strategies.
• It enables setting realistic expectations for growth.
• It supports decision-making regarding investments and expenditures.

Understanding business mathematics allows entrepreneurs to strategize effectively and ensure sustainable growth. This, in turn, can contribute to improved financial health and stability for the business over time.