Problem 96 - College Algebra with Modeling and Visualization

Question

Exercise, Salaries A person has the given starting salary \(S\) and receives a raise \(R\) each year thereafter. (a) Use a formula to calculate the total amount earned over 15 years. (b) Use a calculator to verify this value. $$S-\$ 35,000, R-\$ 2500$$

Step-by-Step Solution

Answer

Total earnings over 15 years is \( \$787,500 \).

Step 1: Understand the Problem

We have a salary increase problem where a person has a starting salary of \( S = \\(35,000 \) and receives a raise of \( R = \\)2,500 \) annually. We need to calculate the total earnings over 15 years.

Step 2: Recognize the Sequence Type

The earnings over 15 years form an arithmetic sequence, where the first term is the initial salary \( S = \\(35,000 \), and the common difference (annual raise) is \( R = \\)2,500 \).

Step 3: Identify the Formula for Arithmetic Series Sum

The sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:\[ S_n = \frac{n}{2} (2a + (n-1)d) \]where \( a = 35000 \) is the first term, \( d = 2500 \) is the common difference, and \( n = 15 \) is the number of terms.

Step 4: Substitute the Values into the Formula

Substitute \( a = 35,000 \), \( d = 2,500 \), and \( n = 15 \) into the sum formula:\[ S_{15} = \frac{15}{2} (2 \times 35,000 + (15-1) \times 2,500) \]

Step 5: Perform Arithmetic Calculations

Calculate the expression: - Calculate \( 2 \times 35,000 = 70,000 \).- Calculate \( (15-1) \times 2,500 = 35,000 \).- Sum these: \( 70,000 + 35,000 = 105,000 \).- Calculate the final sum: \( \frac{15}{2} \times 105,000 = 787,500 \).

Step 6: Verify with a Calculator

Use a calculator to verify each calculation step, ensuring the final total is \( \$787,500 \).

Key Concepts

Arithmetic Series Sum

An arithmetic series is the sum of the terms of an arithmetic sequence. In an arithmetic sequence, each term is generated by adding a constant, known as the common difference, to the previous term. To find the sum of the first \( n \) terms of an arithmetic sequence, you use the formula: \[ S_n = \frac{n}{2} (2a + (n-1)d) \] Here, \( n \) represents the number of terms to sum, \( a \) is the first term, and \( d \) is the common difference between consecutive terms. This formula works by calculating the average of the first and last term and then multiplying by the total number of terms.

• Formula: Saves time and ensures accuracy in calculating sums rapidly.
• Understanding Patterns: Recognizing arithmetic sequences helps identify the series sum effectively.

Breaking down more complex problems into smaller sequences can simplify calculations.

Salary Calculation

In real-world applications, the concept of arithmetic sequences can be used to calculate total earnings over time when salaries increase by a constant amount annually. If a starting salary is given, along with an annual raise, you can construct an arithmetic sequence where:

• First Term: Corresponds to the starting salary.
• Common Difference: Is the annual raise amount.

The sum formula helps compute the total salary over the desired period. For example, if someone starts with a salary of \(\(35,000\) and receives a raise of \(\)2,500\) per year, the total earnings over 15 years can be calculated using the arithmetic series formula.
This method ensures that every raise is taken into account correctly and provides a straightforward tool for financial planning over a fixed time period.

Mathematics Problem Solving

Approaching problem-solving in mathematics involves several strategic steps. Firstly, understanding the problem context is crucial. In our given task, identifying that the problem is an arithmetic sequence is the key step. Then, simplifying the problem using algebraic expressions and known formulas leads to an efficient solution.

• Identify: The type of sequence or pattern involved.
• Formulate: Use appropriate mathematical models or formulas.
• Calculate: Perform accurate arithmetic operations.
• Verify: Double-check calculations for errors using a calculator.

These steps encompass a structured approach to solving mathematical problems, ensuring a clear pathway from understanding to solution verification. Such strategies increase accuracy and build confidence in handling complex problems.

Algebraic Formulas

Algebraic formulas are crucial for solving and simplifying mathematical problems, especially when dealing with arithmetic sequences and series sums. They provide a concise way to express relationships between variables and constants.
The formula for an arithmetic series involves key algebraic components:

• Variables: Represent terms, common differences, and the number of terms.
• Operations: Addition, multiplication, and division help structure the calculation.

Using algebraic formulas minimizes potential errors involved in repeated calculations, ensuring a more reliable solution to problems. Practicing these formulas enhances one's mathematical skills and can be applied in various real-world contexts, from finance to engineering.
Accurate application of these formulas requires a firm understanding of the principles behind them, illustrating the importance of foundational algebra knowledge.