Problem 90 - Precalculus with Limits

Question

In Exercises 89 and 90, consider a job offer with the given starting salary and the given annual raise. (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. Starting Salary \( \$36,800 Annual Raise \) \$ 1750 $

Step-by-Step Solution

Answer

The salary during the sixth year of employment is \$ 45,300. The total compensation from the company through six full years of employment is \$ 257,300.

Step 1: Calculate sixth year salary

To calculate the salary during the sixth year of employment, start with the starting salary and add the annual raise multiplied by the number of years since the initial salary, which in this case is 5. Hence, the salary for the sixth year would be \( \$ 36,800 + ( 5 \times \$ 1,750 )\).

Step 2: Determine total compensation

To determine the total compensation for the full six years, the salaries for each year should be added together. For Year 1, it is the starting salary of \$36,800. For Year 2 to 6, each year's salary can be determined by \( \$36,800 + (n-1) \times \$ 1,750 \), with n representing the year number. Summing all these salaries gives the total compensation.

Step 3: Evaluate the sums

Evaluate the sums calculated in Step 1 for the sixth year salary and Step 2 for total compensation.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term differs from the previous one by a constant amount, known as the common difference. This regularity in progression makes arithmetic sequences one of the simplest types of sequences. They are essential in mathematical applications, especially for problems involving linear growth or decline.

The formula to find any term in an arithmetic sequence is given by:

• General term formula: \( a_n = a_1 + (n-1)d \)

where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term
• \( n \) is the term number
• \( d \) is the common difference

This formula helps pinpoint any term in the sequence when you know the start point, the term you wish to find, and the difference between each term. In the context of salary increments, the starting salary signifies the first term and the annual raise is the common difference.

Salary Calculation

Calculating salaries over a period involves applying the concept of arithmetic sequences. Given a starting salary and an annual raise, each year's salary can be calculated step by step.

For instance, suppose your starting salary is \( \\(36,800 \) and you're offered an annual raise of \( \\)1,750 \). You can calculate the salary during any specific year by using the general term formula of an arithmetic sequence. So, in the sixth year, the salary can be calculated as:

• \( 36,800 + (6-1) \times 1,750 \)
• Which simplifies to \( 36,800 + 8,750 = \$45,550 \)

This ensures you understand the growth of income year on year with each raise factored into the calculations.

Annual Raise

An annual raise refers to the fixed sum added to an employee’s salary every year. This is quite a common practice across various industries to incentivize and reward employees for their contributions. Understanding how it affects net income over time requires a strategic approach to salary calculations.

Let's look at how this works in practice over six years.

• Year 1 salary: Starting at \( \\(36,800 \)
• Year 2 salary: \( 36,800 + 1 \times 1,750 = \\)38,550 \)
• Year 3 salary: \( 36,800 + 2 \times 1,750 = \\(40,300 \)
• Year 4 salary: \( 36,800 + 3 \times 1,750 = \\)42,050 \)
• Year 5 salary: \( 36,800 + 4 \times 1,750 = \\(43,800 \)
• Year 6 salary: \( 36,800 + 5 \times 1,750 = \\)45,550 \)

To calculate the total compensation over six years, add these annual amounts together. The raise not only boosts annual income but also cumulatively enhances overall earning potential over the specified period.